Question 22527: A Complete Initial State Gives a Unique Forward Prefix

Original question and givens

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).

• G001 — 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).

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=3;oldest-to-newest coefficients=[2, 4, 6];constant=-1;ALL initial a_0..a_(r-1)=[1, 2, 3]

for EVERY n>=3,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=3;forward rule is total and single-valued

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 := mul(2, 1)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 = 2

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(-1, 2)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 1

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 := mul(4, 2)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 = 8

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(1, 8)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 9

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 := mul(6, 3)

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 = 18

recurrence ORIGINAL n=3 partial sum after coefficient2 := sum(9, 18)

recurrence ORIGINAL n=3 partial sum after coefficient2 = 27

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

new a_3=27;next state=[2, 3, 27];drop ONLY oldest state entry then append new scalar;prefix now=[1, 2, 3, 27]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 := mul(2, 2)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 = 4

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

recurrence ORIGINAL n=4 partial sum after coefficient0 = 3

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

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 = 12

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(3, 12)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 15

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 := mul(6, 27)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 = 162

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(15, 162)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 177

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[2, 3, 27];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 2, 4], [1, 2, 4, 3, 12], [2, 3, 6, 27, 162]]

new a_4=177;next state=[3, 27, 177];drop ONLY oldest state entry then append new scalar;prefix now=[1, 2, 3, 27, 177]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 := mul(2, 3)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 = 6

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(-1, 6)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 5

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 := mul(4, 27)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 = 108

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(5, 108)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 113

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 := mul(6, 177)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 = 1062

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(113, 1062)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 1175

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[3, 27, 177];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 3, 6], [1, 3, 4, 27, 108], [2, 4, 6, 177, 1062]]

new a_5=1175;next state=[27, 177, 1175];drop ONLY oldest state entry then append new scalar;prefix now=[1, 2, 3, 27, 177, 1175]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 := mul(2, 27)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 = 54

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(-1, 54)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 53

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 := mul(4, 177)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 = 708

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(53, 708)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 761

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 := mul(6, 1175)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 = 7050

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(761, 7050)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 7811

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[27, 177, 1175];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 27, 54], [1, 4, 4, 177, 708], [2, 5, 6, 1175, 7050]]

new a_6=7811;next state=[177, 1175, 7811];drop ONLY oldest state entry then append new scalar;prefix now=[1, 2, 3, 27, 177, 1175, 7811]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 := mul(2, 177)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 = 354

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(-1, 354)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 353

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 := mul(4, 1175)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 = 4700

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(353, 4700)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 5053

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 := mul(6, 7811)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 = 46866

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(5053, 46866)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 51919

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[177, 1175, 7811];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 177, 354], [1, 5, 4, 1175, 4700], [2, 6, 6, 7811, 46866]]

new a_7=51919;next state=[1175, 7811, 51919];drop ONLY oldest state entry then append new scalar;prefix now=[1, 2, 3, 27, 177, 1175, 7811, 51919]

recurrence ORIGINAL:complete ORIGINAL initial state=[1, 2, 3];actual generated prefix=[1, 2, 3, 27, 177, 1175, 7811, 51919]

ANY two sequences with SAME r initials and SAME total rule agree at n=3 by substitution,then at every later n by induction;this determines the requested prefix only

Reverse checks

• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute recurrence ORIGINAL n=5 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 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

(3, 3, 7, 51919, 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.