Question 22550: 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=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).

• G001 — 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).

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=2;ALL initial a_0..a_(r-1)=[24, 25]

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, 24)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 24

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(2, 24)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 26

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(2, 25)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 50

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(26, 50)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 76

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[24, 25];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 24, 24], [1, 1, 2, 25, 50]]

new a_2=76;next state=[25, 76];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 76]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(1, 25)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 25

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(2, 25)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 27

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(2, 76)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 152

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(27, 152)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 179

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[25, 76];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 25, 25], [1, 2, 2, 76, 152]]

new a_3=179;next state=[76, 179];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 76, 179]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(1, 76)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 76

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(2, 76)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 78

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(2, 179)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 358

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(78, 358)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 436

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[76, 179];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 76, 76], [1, 3, 2, 179, 358]]

new a_4=436;next state=[179, 436];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 76, 179, 436]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(1, 179)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 179

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(2, 179)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 181

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(2, 436)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 872

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(181, 872)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 1053

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[179, 436];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 179, 179], [1, 4, 2, 436, 872]]

new a_5=1053;next state=[436, 1053];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 76, 179, 436, 1053]

recurrence ORIGINAL:complete ORIGINAL initial state=[24, 25];actual generated prefix=[24, 25, 76, 179, 436, 1053]

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, 1053, 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.