# Question 22593: An Order-r Total Rule Needs All r Initial Values

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0044-22593 · q-20261014-advance-20261110-r05-recurrence-relation-0018



## Original question and givens

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).


- G001 — 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).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-recurrence-relation](https://ocme.evemisslab.com/en/mko/mko-recurrence-relation/)





## Step-by-step derivation

S001 · impliesBefore

source operation=recurrence-specification-qualification;original snapshot and mathematics retained

After

explicit ordered source output fields=[order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope]

Conditions: none

Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

recurrence ORIGINAL:ORIGINAL order=4;oldest-to-newest coefficients=[2, 4, 6, 8];constant=0;ALL initial a_0..a_(r-1)=[17, 18, 19, 20]

After

for EVERY n>=4,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=4;forward rule is total and single-valued

Conditions: none

Fix all r initials and the oldest-to-newest coefficient roles.Given the preceding r values,the next value is uniquely determined.This is a forward definition,not a closed form or convergence claim.

Step identifiers and dependenciesRule: recurrence_complete_initial_state_and_total_single_value_rule · Uses: G001, S001





S003 · calculationBefore

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 := mul(2, 17)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 34

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(0, 34)

After

recurrence ORIGINAL n=4 partial sum after coefficient0 = 34

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(4, 18)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 72

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(34, 72)

After

recurrence ORIGINAL n=4 partial sum after coefficient1 = 106

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(6, 19)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 114

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(106, 114)

After

recurrence ORIGINAL n=4 partial sum after coefficient2 = 220

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(8, 20)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 160

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(220, 160)

After

recurrence ORIGINAL n=4 partial sum after coefficient3 = 380

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · impliesBefore

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[17, 18, 19, 20];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 17, 34], [1, 1, 4, 18, 72], [2, 2, 6, 19, 114], [3, 3, 8, 20, 160]]

After

new a_4=380;next state=[18, 19, 20, 380];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 19, 20, 380]

Conditions: none

Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update.

Step identifiers and dependenciesRule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S010





S012 · impliesBefore

recurrence ORIGINAL:complete ORIGINAL initial state=[17, 18, 19, 20];actual generated prefix=[17, 18, 19, 20, 380]

After

ANY two sequences with SAME r initials and SAME total rule agree at n=4 by substitution,then at every later n by induction;this determines the requested prefix only

Conditions: none

Induct from equality of all r initials:the same single-valued rule gives equality of the next term,and hence every later prefix term.Finite generation proves no closed form,asymptotic behavior or analyticity.

Step identifiers and dependenciesRule: recurrence_forward_uniqueness_induction_not_analytic_claim · Uses: G001, S011









## Reverse checks


- S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S010 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
- S009 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

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

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 2733198e89792bf65679eff985e862e0111f8a09648b59b1162a65fa36c69765
- candidate record SHA-256: a0d473dc097bd55c0383eef7187cc59b135b5021444185b0a0734e31c352728a
- semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
- question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
- Lean: NOT_RUN · human review: NOT_PERFORMED

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