Question 22538: A Complete Initial State Gives a Unique Forward Prefix

Original question and givens

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).

• G001 — 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).

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

Step-by-step derivation

source operation=recurrence-initial-value-uniqueness;original snapshot and mathematics retained

explicit ordered source output fields=[order, initial_count, target_n, target_value, unique_prefix, certificate]

recurrence ORIGINAL:ORIGINAL order=2;oldest-to-newest coefficients=[1, 2];constant=0;ALL initial a_0..a_(r-1)=[12, 13]

for EVERY n>=2,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=2;forward rule is total and single-valued

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 := mul(1, 12)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 12

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(0, 12)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 12

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(2, 13)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 26

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(12, 26)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 38

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[12, 13];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 12, 12], [1, 1, 2, 13, 26]]

new a_2=38;next state=[13, 38];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(1, 13)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 13

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(0, 13)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 13

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(2, 38)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 76

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(13, 76)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 89

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[13, 38];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 13, 13], [1, 2, 2, 38, 76]]

new a_3=89;next state=[38, 89];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38, 89]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(1, 38)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 38

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(0, 38)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 38

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(2, 89)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 178

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(38, 178)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 216

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[38, 89];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 38, 38], [1, 3, 2, 89, 178]]

new a_4=216;next state=[89, 216];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38, 89, 216]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(1, 89)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 89

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(0, 89)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 89

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(2, 216)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 432

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(89, 432)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 521

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[89, 216];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 89, 89], [1, 4, 2, 216, 432]]

new a_5=521;next state=[216, 521];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38, 89, 216, 521]

recurrence ORIGINAL:complete ORIGINAL initial state=[12, 13];actual generated prefix=[12, 13, 38, 89, 216, 521]

ANY two sequences with SAME r initials and SAME total rule agree at n=2 by substitution,then at every later n by induction;this determines the requested prefix only

Reverse checks

• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute recurrence ORIGINAL n=3 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute recurrence ORIGINAL n=2 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute recurrence ORIGINAL n=2 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute recurrence ORIGINAL n=2 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

(2, 2, 5, 521, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

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.