Question 22547: A Complete Initial State Gives a Unique Forward Prefix

Original question and givens

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).

• G001 — 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).

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=[2, 4];constant=-1;ALL initial a_0..a_(r-1)=[21, 22]

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(2, 21)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 42

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(-1, 42)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 41

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(4, 22)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 88

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(41, 88)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 129

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[21, 22];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 21, 42], [1, 1, 4, 22, 88]]

new a_2=129;next state=[22, 129];drop ONLY oldest state entry then append new scalar;prefix now=[21, 22, 129]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(2, 22)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 44

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(-1, 44)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 43

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(4, 129)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 516

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(43, 516)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 559

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[22, 129];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 22, 44], [1, 2, 4, 129, 516]]

new a_3=559;next state=[129, 559];drop ONLY oldest state entry then append new scalar;prefix now=[21, 22, 129, 559]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(2, 129)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 258

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(-1, 258)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 257

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(4, 559)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 2236

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(257, 2236)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 2493

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[129, 559];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 129, 258], [1, 3, 4, 559, 2236]]

new a_4=2493;next state=[559, 2493];drop ONLY oldest state entry then append new scalar;prefix now=[21, 22, 129, 559, 2493]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(2, 559)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 1118

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(-1, 1118)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 1117

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(4, 2493)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 9972

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(1117, 9972)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 11089

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[559, 2493];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 559, 1118], [1, 4, 4, 2493, 9972]]

new a_5=11089;next state=[2493, 11089];drop ONLY oldest state entry then append new scalar;prefix now=[21, 22, 129, 559, 2493, 11089]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 := mul(2, 2493)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 = 4986

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(-1, 4986)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 4985

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 := mul(4, 11089)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 = 44356

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(4985, 44356)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 49341

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[2493, 11089];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 2493, 4986], [1, 5, 4, 11089, 44356]]

new a_6=49341;next state=[11089, 49341];drop ONLY oldest state entry then append new scalar;prefix now=[21, 22, 129, 559, 2493, 11089, 49341]

recurrence ORIGINAL:complete ORIGINAL initial state=[21, 22];actual generated prefix=[21, 22, 129, 559, 2493, 11089, 49341]

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

• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 with exact rational arithmetic, not rounded decimals.
• 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, 6, 49341, 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.