# Oldest-to-Newest State Updates, Complete Initials and Unique Prefixes

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261110-r05-recurrence-relation-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. An Order-r Total Rule Needs All r Initial Values

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-2 and initial_values=[0, 1], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(2, [1, 2], -2, [0, 1], Yes, Yes, rule_applies_for_n_ge_2)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [1, 2], -2, [0, 1], Yes, Yes, rule_applies_for_n_ge_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0001/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 2. An Order-r Total Rule Needs All r Initial Values

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=-1 and initial_values=[1, 2, 3], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(3, [2, 4, 6], -1, [1, 2, 3], Yes, Yes, rule_applies_for_n_ge_3)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [2, 4, 6], -1, [1, 2, 3], Yes, Yes, rule_applies_for_n_ge_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0002/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 3. An Order-r Total Rule Needs All r Initial Values

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=0 and initial_values=[2, 3, 4, 5], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(4, [1, 2, 3, 4], 0, [2, 3, 4, 5], Yes, Yes, rule_applies_for_n_ge_4)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], 0, [2, 3, 4, 5], Yes, Yes, rule_applies_for_n_ge_4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0003/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 4. An Order-r Total Rule Needs All r Initial Values

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=1 and initial_values=[3, 4], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(2, [2, 4], 1, [3, 4], Yes, Yes, rule_applies_for_n_ge_2)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [2, 4], 1, [3, 4], Yes, Yes, rule_applies_for_n_ge_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0004/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 5. An Order-r Total Rule Needs All r Initial Values

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=2 and initial_values=[4, 5, 6], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(3, [1, 2, 3], 2, [4, 5, 6], Yes, Yes, rule_applies_for_n_ge_3)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [1, 2, 3], 2, [4, 5, 6], Yes, Yes, rule_applies_for_n_ge_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0005/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 6. An Order-r Total Rule Needs All r Initial Values

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-2 and initial_values=[5, 6, 7, 8], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(4, [2, 4, 6, 8], -2, [5, 6, 7, 8], Yes, Yes, rule_applies_for_n_ge_4)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [2, 4, 6, 8], -2, [5, 6, 7, 8], Yes, Yes, rule_applies_for_n_ge_4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0006/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 7. An Order-r Total Rule Needs All r Initial Values

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-1 and initial_values=[6, 7], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(2, [1, 2], -1, [6, 7], Yes, Yes, rule_applies_for_n_ge_2)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [1, 2], -1, [6, 7], Yes, Yes, rule_applies_for_n_ge_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0007/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 8. An Order-r Total Rule Needs All r Initial Values

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=0 and initial_values=[7, 8, 9], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(3, [2, 4, 6], 0, [7, 8, 9], Yes, Yes, rule_applies_for_n_ge_3)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [2, 4, 6], 0, [7, 8, 9], Yes, Yes, rule_applies_for_n_ge_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0008/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 9. An Order-r Total Rule Needs All r Initial Values

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=1 and initial_values=[8, 9, 10, 11], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(4, [1, 2, 3, 4], 1, [8, 9, 10, 11], Yes, Yes, rule_applies_for_n_ge_4)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], 1, [8, 9, 10, 11], Yes, Yes, rule_applies_for_n_ge_4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0009/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 10. An Order-r Total Rule Needs All r Initial Values

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=2 and initial_values=[9, 10], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(2, [2, 4], 2, [9, 10], Yes, Yes, rule_applies_for_n_ge_2)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [2, 4], 2, [9, 10], Yes, Yes, rule_applies_for_n_ge_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0010/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 11. An Order-r Total Rule Needs All r Initial Values

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-2 and initial_values=[10, 11, 12], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(3, [1, 2, 3], -2, [10, 11, 12], Yes, Yes, rule_applies_for_n_ge_3)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [1, 2, 3], -2, [10, 11, 12], Yes, Yes, rule_applies_for_n_ge_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0011/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 12. An Order-r Total Rule Needs All r Initial Values

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-1 and initial_values=[11, 12, 13, 14], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(4, [2, 4, 6, 8], -1, [11, 12, 13, 14], Yes, Yes, rule_applies_for_n_ge_4)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [2, 4, 6, 8], -1, [11, 12, 13, 14], Yes, Yes, rule_applies_for_n_ge_4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0012/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 13. An Order-r Total Rule Needs All r Initial Values

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=0 and initial_values=[12, 13], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(2, [1, 2], 0, [12, 13], Yes, Yes, rule_applies_for_n_ge_2)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [1, 2], 0, [12, 13], Yes, Yes, rule_applies_for_n_ge_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0013/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 14. An Order-r Total Rule Needs All r Initial Values

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=1 and initial_values=[13, 14, 15], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(3, [2, 4, 6], 1, [13, 14, 15], Yes, Yes, rule_applies_for_n_ge_3)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [2, 4, 6], 1, [13, 14, 15], Yes, Yes, rule_applies_for_n_ge_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0014/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 15. An Order-r Total Rule Needs All r Initial Values

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=2 and initial_values=[14, 15, 16, 17], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(4, [1, 2, 3, 4], 2, [14, 15, 16, 17], Yes, Yes, rule_applies_for_n_ge_4)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], 2, [14, 15, 16, 17], Yes, Yes, rule_applies_for_n_ge_4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0015/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 16. An Order-r Total Rule Needs All r Initial Values

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-2 and initial_values=[15, 16], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(2, [2, 4], -2, [15, 16], Yes, Yes, rule_applies_for_n_ge_2)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [2, 4], -2, [15, 16], Yes, Yes, rule_applies_for_n_ge_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0016/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 17. An Order-r Total Rule Needs All r Initial Values

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-1 and initial_values=[16, 17, 18], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(3, [1, 2, 3], -1, [16, 17, 18], Yes, Yes, rule_applies_for_n_ge_3)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [1, 2, 3], -1, [16, 17, 18], Yes, Yes, rule_applies_for_n_ge_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0017/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 18. An Order-r Total Rule Needs All r Initial Values

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=0 and initial_values=[17, 18, 19, 20], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(4, [2, 4, 6, 8], 0, [17, 18, 19, 20], Yes, Yes, rule_applies_for_n_ge_4)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [2, 4, 6, 8], 0, [17, 18, 19, 20], Yes, Yes, rule_applies_for_n_ge_4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0018/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 19. An Order-r Total Rule Needs All r Initial Values

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=1 and initial_values=[18, 19], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(2, [1, 2], 1, [18, 19], Yes, Yes, rule_applies_for_n_ge_2)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [1, 2], 1, [18, 19], Yes, Yes, rule_applies_for_n_ge_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0019/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 20. An Order-r Total Rule Needs All r Initial Values

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=2 and initial_values=[19, 20, 21], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(3, [2, 4, 6], 2, [19, 20, 21], Yes, Yes, rule_applies_for_n_ge_3)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [2, 4, 6], 2, [19, 20, 21], Yes, Yes, rule_applies_for_n_ge_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0020/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 21. An Order-r Total Rule Needs All r Initial Values

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=-2 and initial_values=[20, 21, 22, 23], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(4, [1, 2, 3, 4], -2, [20, 21, 22, 23], Yes, Yes, rule_applies_for_n_ge_4)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], -2, [20, 21, 22, 23], Yes, Yes, rule_applies_for_n_ge_4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0021/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 22. An Order-r Total Rule Needs All r Initial Values

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-1 and initial_values=[21, 22], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(2, [2, 4], -1, [21, 22], Yes, Yes, rule_applies_for_n_ge_2)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [2, 4], -1, [21, 22], Yes, Yes, rule_applies_for_n_ge_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0022/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 23. An Order-r Total Rule Needs All r Initial Values

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=0 and initial_values=[22, 23, 24], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(3, [1, 2, 3], 0, [22, 23, 24], Yes, Yes, rule_applies_for_n_ge_3)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [1, 2, 3], 0, [22, 23, 24], Yes, Yes, rule_applies_for_n_ge_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0023/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 24. An Order-r Total Rule Needs All r Initial Values

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=1 and initial_values=[23, 24, 25, 26], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(4, [2, 4, 6, 8], 1, [23, 24, 25, 26], Yes, Yes, rule_applies_for_n_ge_4)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [2, 4, 6, 8], 1, [23, 24, 25, 26], Yes, Yes, rule_applies_for_n_ge_4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0024/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 25. An Order-r Total Rule Needs All r Initial Values

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=2 and initial_values=[24, 25], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).

View answer and explanation(2, [1, 2], 2, [24, 25], Yes, Yes, rule_applies_for_n_ge_2)

Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [1, 2], 2, [24, 25], Yes, Yes, rule_applies_for_n_ge_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0025/)

Recurrence relation · qf-mod41-recurrence-specification-qualification · candidate





## 26. Shift the Full State and Append Each Exact New Scalar

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-2 and initial_values=[0, 1], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(2, [0, 1], 5, [0, 1, 0, -1, -4, -11], -11, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [0, 1], 5, [0, 1, 0, -1, -4, -11], -11, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0026/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 27. Shift the Full State and Append Each Exact New Scalar

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=-1 and initial_values=[1, 2, 3], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(3, [1, 2, 3], 7, [1, 2, 3, 27, 177, 1175, 7811, 51919], 51919, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [1, 2, 3], 7, [1, 2, 3, 27, 177, 1175, 7811, 51919], 51919, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0027/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 28. Shift the Full State and Append Each Exact New Scalar

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=0 and initial_values=[2, 3, 4, 5], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(4, [2, 3, 4, 5], 9, [2, 3, 4, 5, 40, 186, 878, 4155, 19666, 93071], 93071, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [2, 3, 4, 5], 9, [2, 3, 4, 5, 40, 186, 878, 4155, 19666, 93071], 93071, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0028/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 29. Shift the Full State and Append Each Exact New Scalar

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=1 and initial_values=[3, 4], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(2, [3, 4], 8, [3, 4, 23, 101, 451, 2007, 8931, 39739, 176819], 176819, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [3, 4], 8, [3, 4, 23, 101, 451, 2007, 8931, 39739, 176819], 176819, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0029/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 30. Shift the Full State and Append Each Exact New Scalar

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=2 and initial_values=[4, 5, 6], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(3, [4, 5, 6], 6, [4, 5, 6, 34, 121, 439, 1595], 1595, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [4, 5, 6], 6, [4, 5, 6, 34, 121, 439, 1595], 1595, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0030/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 31. Shift the Full State and Append Each Exact New Scalar

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-2 and initial_values=[5, 6, 7, 8], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(4, [5, 6, 7, 8], 8, [5, 6, 7, 8, 138, 1190, 10392, 90842, 794122], 794122, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [5, 6, 7, 8], 8, [5, 6, 7, 8, 138, 1190, 10392, 90842, 794122], 794122, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0031/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 32. Shift the Full State and Append Each Exact New Scalar

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-1 and initial_values=[6, 7], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(2, [6, 7], 7, [6, 7, 19, 44, 106, 255, 615, 1484], 1484, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [6, 7], 7, [6, 7, 19, 44, 106, 255, 615, 1484], 1484, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0032/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 33. Shift the Full State and Append Each Exact New Scalar

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=0 and initial_values=[7, 8, 9], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(3, [7, 8, 9], 9, [7, 8, 9, 100, 652, 4330, 28788, 191352, 1271924, 8454528], 8454528, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [7, 8, 9], 9, [7, 8, 9, 100, 652, 4330, 28788, 191352, 1271924, 8454528], 8454528, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0033/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 34. Shift the Full State and Append Each Exact New Scalar

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=1 and initial_values=[8, 9, 10, 11], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(4, [8, 9, 10, 11], 7, [8, 9, 10, 11, 101, 467, 2204, 10431], 10431, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [8, 9, 10, 11], 7, [8, 9, 10, 11, 101, 467, 2204, 10431], 10431, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0034/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 35. Shift the Full State and Append Each Exact New Scalar

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=2 and initial_values=[9, 10], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(2, [9, 10], 6, [9, 10, 60, 262, 1170, 5206, 23166], 23166, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [9, 10], 6, [9, 10, 60, 262, 1170, 5206, 23166], 23166, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0035/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 36. Shift the Full State and Append Each Exact New Scalar

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-2 and initial_values=[10, 11, 12], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(3, [10, 11, 12], 8, [10, 11, 12, 66, 231, 835, 3031, 10992, 39871], 39871, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [10, 11, 12], 8, [10, 11, 12, 66, 231, 835, 3031, 10992, 39871], 39871, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0036/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 37. Shift the Full State and Append Each Exact New Scalar

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-1 and initial_values=[11, 12, 13, 14], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=10. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(4, [11, 12, 13, 14], 10, [11, 12, 13, 14, 259, 2231, 19483, 170313, 1488843, 13015015, 113773395], 113773395, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [11, 12, 13, 14], 10, [11, 12, 13, 14, 259, 2231, 19483, 170313, 1488843, 13015015, 113773395], 113773395, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0037/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 38. Shift the Full State and Append Each Exact New Scalar

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=0 and initial_values=[12, 13], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(2, [12, 13], 5, [12, 13, 38, 89, 216, 521], 521, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [12, 13], 5, [12, 13, 38, 89, 216, 521], 521, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0038/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 39. Shift the Full State and Append Each Exact New Scalar

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=1 and initial_values=[13, 14, 15], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(3, [13, 14, 15], 7, [13, 14, 15, 173, 1127, 7485, 49765, 330785], 330785, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [13, 14, 15], 7, [13, 14, 15, 173, 1127, 7485, 49765, 330785], 330785, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0039/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 40. Shift the Full State and Append Each Exact New Scalar

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=2 and initial_values=[14, 15, 16, 17], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(4, [14, 15, 16, 17], 9, [14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243], 374243, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [14, 15, 16, 17], 9, [14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243], 374243, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0040/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 41. Shift the Full State and Append Each Exact New Scalar

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-2 and initial_values=[15, 16], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(2, [15, 16], 8, [15, 16, 92, 398, 1774, 7890, 35106, 156202, 695018], 695018, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [15, 16], 8, [15, 16, 92, 398, 1774, 7890, 35106, 156202, 695018], 695018, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0041/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 42. Shift the Full State and Append Each Exact New Scalar

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-1 and initial_values=[16, 17, 18], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(3, [16, 17, 18], 6, [16, 17, 18, 103, 361, 1306, 4742], 4742, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [16, 17, 18], 6, [16, 17, 18, 103, 361, 1306, 4742], 4742, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0042/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 43. Shift the Full State and Append Each Exact New Scalar

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=0 and initial_values=[17, 18, 19, 20], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(4, [17, 18, 19, 20], 8, [17, 18, 19, 20, 380, 3272, 28574, 249784, 2183564], 2183564, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [17, 18, 19, 20], 8, [17, 18, 19, 20, 380, 3272, 28574, 249784, 2183564], 2183564, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0043/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 44. Shift the Full State and Append Each Exact New Scalar

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=1 and initial_values=[18, 19], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(2, [18, 19], 7, [18, 19, 57, 134, 326, 787, 1901, 4590], 4590, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [18, 19], 7, [18, 19, 57, 134, 326, 787, 1901, 4590], 4590, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0044/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 45. Shift the Full State and Append Each Exact New Scalar

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=2 and initial_values=[19, 20, 21], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(3, [19, 20, 21], 9, [19, 20, 21, 246, 1602, 10640, 70742, 470218, 3125558, 20775706], 20775706, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [19, 20, 21], 9, [19, 20, 21, 246, 1602, 10640, 70742, 470218, 3125558, 20775706], 20775706, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0045/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 46. Shift the Full State and Append Each Exact New Scalar

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=-2 and initial_values=[20, 21, 22, 23], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(4, [20, 21, 22, 23], 7, [20, 21, 22, 23, 218, 1004, 4736, 22413], 22413, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [20, 21, 22, 23], 7, [20, 21, 22, 23, 218, 1004, 4736, 22413], 22413, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0046/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 47. Shift the Full State and Append Each Exact New Scalar

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-1 and initial_values=[21, 22], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(2, [21, 22], 6, [21, 22, 129, 559, 2493, 11089, 49341], 49341, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [21, 22], 6, [21, 22, 129, 559, 2493, 11089, 49341], 49341, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0047/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 48. Shift the Full State and Append Each Exact New Scalar

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=0 and initial_values=[22, 23, 24], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(3, [22, 23, 24], 8, [22, 23, 24, 140, 491, 1777, 6453, 23404, 84895], 84895, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [22, 23, 24], 8, [22, 23, 24, 140, 491, 1777, 6453, 23404, 84895], 84895, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0048/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 49. Shift the Full State and Append Each Exact New Scalar

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=1 and initial_values=[23, 24, 25, 26], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=10. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(4, [23, 24, 25, 26], 10, [23, 24, 25, 26, 501, 4313, 37665, 329255, 2878285, 25161097, 219950837], 219950837, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [23, 24, 25, 26], 10, [23, 24, 25, 26, 501, 4313, 37665, 329255, 2878285, 25161097, 219950837], 219950837, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0049/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 50. Shift the Full State and Append Each Exact New Scalar

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=2 and initial_values=[24, 25], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_values, target_n, prefix, target_value, certificate).

View answer and explanation(2, [24, 25], 5, [24, 25, 76, 179, 436, 1053], 1053, exact_forward_application_only)

Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [24, 25], 5, [24, 25, 76, 179, 436, 1053], 1053, exact_forward_application_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0050/)

Recurrence relation · qf-mod41-recurrence-forward-generation · candidate





## 51. A Complete Initial State Gives a Unique Forward Prefix

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-2 and initial_values=[0, 1], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(2, 2, 5, -11, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 5, -11, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0051/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 52. A Complete Initial State Gives a Unique Forward Prefix

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=-1 and initial_values=[1, 2, 3], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(3, 3, 7, 51919, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 7, 51919, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0052/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 53. A Complete Initial State Gives a Unique Forward Prefix

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=0 and initial_values=[2, 3, 4, 5], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(4, 4, 9, 93071, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 9, 93071, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0053/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 54. A Complete Initial State Gives a Unique Forward Prefix

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=1 and initial_values=[3, 4], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(2, 2, 8, 176819, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 8, 176819, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0054/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 55. A Complete Initial State Gives a Unique Forward Prefix

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=2 and initial_values=[4, 5, 6], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(3, 3, 6, 1595, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 6, 1595, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0055/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 56. A Complete Initial State Gives a Unique Forward Prefix

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-2 and initial_values=[5, 6, 7, 8], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(4, 4, 8, 794122, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 8, 794122, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0056/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 57. A Complete Initial State Gives a Unique Forward Prefix

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-1 and initial_values=[6, 7], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(2, 2, 7, 1484, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 7, 1484, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0057/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 58. A Complete Initial State Gives a Unique Forward Prefix

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=0 and initial_values=[7, 8, 9], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(3, 3, 9, 8454528, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 9, 8454528, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0058/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 59. A Complete Initial State Gives a Unique Forward Prefix

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=1 and initial_values=[8, 9, 10, 11], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(4, 4, 7, 10431, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 7, 10431, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0059/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 60. A Complete Initial State Gives a Unique Forward Prefix

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=2 and initial_values=[9, 10], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(2, 2, 6, 23166, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 6, 23166, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0060/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 61. A Complete Initial State Gives a Unique Forward Prefix

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-2 and initial_values=[10, 11, 12], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(3, 3, 8, 39871, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 8, 39871, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0061/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 62. A Complete Initial State Gives a Unique Forward Prefix

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-1 and initial_values=[11, 12, 13, 14], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=10. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(4, 4, 10, 113773395, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 10, 113773395, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0062/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 63. A Complete Initial State Gives a Unique Forward Prefix

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=0 and initial_values=[12, 13], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(2, 2, 5, 521, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 5, 521, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0063/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 64. A Complete Initial State Gives a Unique Forward Prefix

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=1 and initial_values=[13, 14, 15], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(3, 3, 7, 330785, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 7, 330785, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0064/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 65. A Complete Initial State Gives a Unique Forward Prefix

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=2 and initial_values=[14, 15, 16, 17], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(4, 4, 9, 374243, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 9, 374243, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0065/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 66. A Complete Initial State Gives a Unique Forward Prefix

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-2 and initial_values=[15, 16], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(2, 2, 8, 695018, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 8, 695018, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0066/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 67. A Complete Initial State Gives a Unique Forward Prefix

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-1 and initial_values=[16, 17, 18], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(3, 3, 6, 4742, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 6, 4742, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0067/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 68. A Complete Initial State Gives a Unique Forward Prefix

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=0 and initial_values=[17, 18, 19, 20], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(4, 4, 8, 2183564, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 8, 2183564, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0068/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 69. A Complete Initial State Gives a Unique Forward Prefix

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=1 and initial_values=[18, 19], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(2, 2, 7, 4590, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 7, 4590, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0069/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 70. A Complete Initial State Gives a Unique Forward Prefix

Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=2 and initial_values=[19, 20, 21], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(3, 3, 9, 20775706, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 9, 20775706, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0070/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 71. A Complete Initial State Gives a Unique Forward Prefix

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=-2 and initial_values=[20, 21, 22, 23], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(4, 4, 7, 22413, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 7, 22413, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0071/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 72. A Complete Initial State Gives a Unique Forward Prefix

Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-1 and initial_values=[21, 22], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(2, 2, 6, 49341, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 6, 49341, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0072/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 73. A Complete Initial State Gives a Unique Forward Prefix

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=0 and initial_values=[22, 23, 24], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(3, 3, 8, 84895, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 8, 84895, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0073/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 74. A Complete Initial State Gives a Unique Forward Prefix

Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=1 and initial_values=[23, 24, 25, 26], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=10. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(4, 4, 10, 219950837, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 10, 219950837, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0074/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 75. A Complete Initial State Gives a Unique Forward Prefix

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=2 and initial_values=[24, 25], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

View answer and explanation(2, 2, 5, 1053, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 5, 1053, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0075/)

Recurrence relation · qf-mod41-recurrence-initial-value-uniqueness · candidate





## 76. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[1, 2] oldest-to-newest, constant=0, known a_0=0, alternative a_1=1 or 2, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(0, 1, 2, [0, 1, 2, 5, 12, 29, 70], [0, 2, 4, 10, 24, 58, 140], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (0, 1, 2, [0, 1, 2, 5, 12, 29, 70], [0, 2, 4, 10, 24, 58, 140], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0076/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 77. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[2, 3] oldest-to-newest, constant=1, known a_0=1, alternative a_1=2 or 3, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(1, 2, 3, [1, 2, 9, 32, 115, 410, 1461, 5204], [1, 3, 12, 43, 154, 549, 1956, 6967], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (1, 2, 3, [1, 2, 9, 32, 115, 410, 1461, 5204], [1, 3, 12, 43, 154, 549, 1956, 6967], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0077/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 78. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[3, 4] oldest-to-newest, constant=2, known a_0=2, alternative a_1=3 or 4, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(2, 3, 4, [2, 3, 20, 91, 426, 1979, 9196, 42723, 198482], [2, 4, 24, 110, 514, 2388, 11096, 51550, 239490], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (2, 3, 4, [2, 3, 20, 91, 426, 1979, 9196, 42723, 198482], [2, 4, 24, 110, 514, 2388, 11096, 51550, 239490], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0078/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 79. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[1, 5] oldest-to-newest, constant=3, known a_0=3, alternative a_1=4 or 5, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(3, 4, 5, [3, 4, 26, 137, 714, 3710, 19267], [3, 5, 31, 163, 849, 4411, 22907], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (3, 4, 5, [3, 4, 26, 137, 714, 3710, 19267], [3, 5, 31, 163, 849, 4411, 22907], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0079/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 80. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[2, 6] oldest-to-newest, constant=0, known a_0=4, alternative a_1=5 or 6, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(4, 5, 6, [4, 5, 38, 238, 1504, 9500, 60008, 379048], [4, 6, 44, 276, 1744, 11016, 69584, 439536], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (4, 5, 6, [4, 5, 38, 238, 1504, 9500, 60008, 379048], [4, 6, 44, 276, 1744, 11016, 69584, 439536], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0080/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 81. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[3, 2] oldest-to-newest, constant=1, known a_0=5, alternative a_1=6 or 7, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(5, 6, 7, [5, 6, 28, 75, 235, 696, 2098, 6285, 18865], [5, 7, 30, 82, 255, 757, 2280, 6832, 20505], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (5, 6, 7, [5, 6, 28, 75, 235, 696, 2098, 6285, 18865], [5, 7, 30, 82, 255, 757, 2280, 6832, 20505], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0081/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 82. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[1, 3] oldest-to-newest, constant=2, known a_0=6, alternative a_1=7 or 8, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(6, 7, 8, [6, 7, 29, 96, 319, 1055, 3486], [6, 8, 32, 106, 352, 1164, 3846], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (6, 7, 8, [6, 7, 29, 96, 319, 1055, 3486], [6, 8, 32, 106, 352, 1164, 3846], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0082/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 83. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[2, 4] oldest-to-newest, constant=3, known a_0=7, alternative a_1=8 or 9, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(7, 8, 9, [7, 8, 49, 215, 961, 4277, 19033, 84689], [7, 9, 53, 233, 1041, 4633, 20617, 91737], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (7, 8, 9, [7, 8, 49, 215, 961, 4277, 19033, 84689], [7, 9, 53, 233, 1041, 4633, 20617, 91737], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0083/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 84. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[3, 5] oldest-to-newest, constant=0, known a_0=8, alternative a_1=9 or 10, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(8, 9, 10, [8, 9, 69, 372, 2067, 11451, 63456, 351633, 1948533], [8, 10, 74, 400, 2222, 12310, 68216, 378010, 2094698], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (8, 9, 10, [8, 9, 69, 372, 2067, 11451, 63456, 351633, 1948533], [8, 10, 74, 400, 2222, 12310, 68216, 378010, 2094698], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0084/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 85. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[1, 6] oldest-to-newest, constant=1, known a_0=9, alternative a_1=10 or 11, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(9, 10, 11, [9, 10, 70, 431, 2657, 16374, 100902], [9, 11, 76, 468, 2885, 17779, 109560], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (9, 10, 11, [9, 10, 70, 431, 2657, 16374, 100902], [9, 11, 76, 468, 2885, 17779, 109560], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0085/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 86. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[2, 2] oldest-to-newest, constant=2, known a_0=10, alternative a_1=11 or 12, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(10, 11, 12, [10, 11, 44, 112, 314, 854, 2338, 6386], [10, 12, 46, 118, 330, 898, 2458, 6714], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (10, 11, 12, [10, 11, 44, 112, 314, 854, 2338, 6386], [10, 12, 46, 118, 330, 898, 2458, 6714], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0086/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 87. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[3, 3] oldest-to-newest, constant=3, known a_0=11, alternative a_1=12 or 13, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(11, 12, 13, [11, 12, 72, 255, 984, 3720, 14115, 53508, 202872], [11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (11, 12, 13, [11, 12, 72, 255, 984, 3720, 14115, 53508, 202872], [11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0087/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 88. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[1, 4] oldest-to-newest, constant=0, known a_0=12, alternative a_1=13 or 14, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(12, 13, 14, [12, 13, 64, 269, 1140, 4829, 20456], [12, 14, 68, 286, 1212, 5134, 21748], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (12, 13, 14, [12, 13, 64, 269, 1140, 4829, 20456], [12, 14, 68, 286, 1212, 5134, 21748], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0088/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 89. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[2, 5] oldest-to-newest, constant=1, known a_0=13, alternative a_1=14 or 15, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(13, 14, 15, [13, 14, 97, 514, 2765, 14854, 79801, 428714], [13, 15, 102, 541, 2910, 15633, 83986, 451197], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (13, 14, 15, [13, 14, 97, 514, 2765, 14854, 79801, 428714], [13, 15, 102, 541, 2910, 15633, 83986, 451197], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0089/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 90. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[3, 6] oldest-to-newest, constant=2, known a_0=14, alternative a_1=15 or 16, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(14, 15, 16, [14, 15, 134, 851, 5510, 35615, 230222, 1488179, 9619742], [14, 16, 140, 890, 5762, 37244, 240752, 1556246, 10059734], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (14, 15, 16, [14, 15, 134, 851, 5510, 35615, 230222, 1488179, 9619742], [14, 16, 140, 890, 5762, 37244, 240752, 1556246, 10059734], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0090/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 91. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[1, 2] oldest-to-newest, constant=3, known a_0=15, alternative a_1=16 or 17, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(15, 16, 17, [15, 16, 50, 119, 291, 704, 1702], [15, 17, 52, 124, 303, 733, 1772], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (15, 16, 17, [15, 16, 50, 119, 291, 704, 1702], [15, 17, 52, 124, 303, 733, 1772], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0091/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 92. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[2, 3] oldest-to-newest, constant=0, known a_0=16, alternative a_1=17 or 18, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(16, 17, 18, [16, 17, 83, 283, 1015, 3611, 12863, 45811], [16, 18, 86, 294, 1054, 3750, 13358, 47574], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (16, 17, 18, [16, 17, 83, 283, 1015, 3611, 12863, 45811], [16, 18, 86, 294, 1054, 3750, 13358, 47574], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0092/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 93. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[3, 4] oldest-to-newest, constant=1, known a_0=17, alternative a_1=18 or 19, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(17, 18, 19, [17, 18, 124, 551, 2577, 11962, 55580, 258207, 1199569], [17, 19, 128, 570, 2665, 12371, 57480, 267034, 1240577], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (17, 18, 19, [17, 18, 124, 551, 2577, 11962, 55580, 258207, 1199569], [17, 19, 128, 570, 2665, 12371, 57480, 267034, 1240577], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0093/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 94. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[1, 5] oldest-to-newest, constant=2, known a_0=18, alternative a_1=19 or 20, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(18, 19, 20, [18, 19, 115, 596, 3097, 16083, 83514], [18, 20, 120, 622, 3232, 16784, 87154], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (18, 19, 20, [18, 19, 115, 596, 3097, 16083, 83514], [18, 20, 120, 622, 3232, 16784, 87154], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0094/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 95. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[2, 6] oldest-to-newest, constant=3, known a_0=19, alternative a_1=20 or 21, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(19, 20, 21, [19, 20, 161, 1009, 6379, 40295, 254531, 1607779], [19, 21, 167, 1047, 6619, 41811, 264107, 1668267], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (19, 20, 21, [19, 20, 161, 1009, 6379, 40295, 254531, 1607779], [19, 21, 167, 1047, 6619, 41811, 264107, 1668267], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0095/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 96. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[3, 2] oldest-to-newest, constant=0, known a_0=20, alternative a_1=21 or 22, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(20, 21, 22, [20, 21, 102, 267, 840, 2481, 7482, 22407, 67260], [20, 22, 104, 274, 860, 2542, 7664, 22954, 68900], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (20, 21, 22, [20, 21, 102, 267, 840, 2481, 7482, 22407, 67260], [20, 22, 104, 274, 860, 2542, 7664, 22954, 68900], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0096/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 97. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[1, 3] oldest-to-newest, constant=1, known a_0=21, alternative a_1=22 or 23, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(21, 22, 23, [21, 22, 88, 287, 950, 3138, 10365], [21, 23, 91, 297, 983, 3247, 10725], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (21, 22, 23, [21, 22, 88, 287, 950, 3138, 10365], [21, 23, 91, 297, 983, 3247, 10725], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0097/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 98. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[2, 4] oldest-to-newest, constant=2, known a_0=22, alternative a_1=23 or 24, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(22, 23, 24, [22, 23, 138, 600, 2678, 11914, 53014, 235886], [22, 24, 142, 618, 2758, 12270, 54598, 242934], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (22, 23, 24, [22, 23, 138, 600, 2678, 11914, 53014, 235886], [22, 24, 142, 618, 2758, 12270, 54598, 242934], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0098/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 99. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[3, 5] oldest-to-newest, constant=3, known a_0=23, alternative a_1=24 or 25, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(23, 24, 25, [23, 24, 192, 1035, 5754, 31878, 176655, 978912, 5424528], [23, 25, 197, 1063, 5909, 32737, 181415, 1005289, 5570693], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (23, 24, 25, [23, 24, 192, 1035, 5754, 31878, 176655, 978912, 5424528], [23, 25, 197, 1063, 5909, 32737, 181415, 1005289, 5570693], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0099/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate





## 100. Different Missing-Initial Completions Give Different Whole Prefixes

Use order2 coefficients=[1, 6] oldest-to-newest, constant=0, known a_0=24, alternative a_1=25 or 26, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different.

View answer and explanation(24, 25, 26, [24, 25, 174, 1069, 6588, 40597, 250170], [24, 26, 180, 1106, 6816, 42002, 258828], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (24, 25, 26, [24, 25, 174, 1069, 6588, 40597, 250170], [24, 26, 180, 1106, 6816, 42002, 258828], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r05-recurrence-relation-0100/)

Recurrence relation · qf-mod41-recurrence-missing-initial-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261110-r05-recurrence-relation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261110-r05-recurrence-relation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261110-r05-recurrence-relation-foundations/validation-report.json)