Question 22542: A Complete Initial State Gives a Unique Forward Prefix

Original question and givens

Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-1 and initial_values=[16, 17, 18], 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=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-1 and initial_values=[16, 17, 18], 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=3;oldest-to-newest coefficients=[1, 2, 3];constant=-1;ALL initial a_0..a_(r-1)=[16, 17, 18]

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(1, 16)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 = 16

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(-1, 16)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 15

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 := mul(2, 17)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 = 34

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(15, 34)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 49

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 := mul(3, 18)

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 = 54

recurrence ORIGINAL n=3 partial sum after coefficient2 := sum(49, 54)

recurrence ORIGINAL n=3 partial sum after coefficient2 = 103

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[16, 17, 18];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 16, 16], [1, 1, 2, 17, 34], [2, 2, 3, 18, 54]]

new a_3=103;next state=[17, 18, 103];drop ONLY oldest state entry then append new scalar;prefix now=[16, 17, 18, 103]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 := mul(1, 17)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 = 17

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(-1, 17)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 16

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 := mul(2, 18)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 = 36

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(16, 36)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 52

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 := mul(3, 103)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 = 309

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(52, 309)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 361

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[17, 18, 103];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 17, 17], [1, 2, 2, 18, 36], [2, 3, 3, 103, 309]]

new a_4=361;next state=[18, 103, 361];drop ONLY oldest state entry then append new scalar;prefix now=[16, 17, 18, 103, 361]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 := mul(1, 18)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 = 18

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(-1, 18)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 17

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 := mul(2, 103)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 = 206

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(17, 206)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 223

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 := mul(3, 361)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 = 1083

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(223, 1083)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 1306

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[18, 103, 361];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 18, 18], [1, 3, 2, 103, 206], [2, 4, 3, 361, 1083]]

new a_5=1306;next state=[103, 361, 1306];drop ONLY oldest state entry then append new scalar;prefix now=[16, 17, 18, 103, 361, 1306]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 := mul(1, 103)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 = 103

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(-1, 103)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 102

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 := mul(2, 361)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 = 722

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(102, 722)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 824

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 := mul(3, 1306)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 = 3918

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(824, 3918)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 4742

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[103, 361, 1306];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 103, 103], [1, 4, 2, 361, 722], [2, 5, 3, 1306, 3918]]

new a_6=4742;next state=[361, 1306, 4742];drop ONLY oldest state entry then append new scalar;prefix now=[16, 17, 18, 103, 361, 1306, 4742]

recurrence ORIGINAL:complete ORIGINAL initial state=[16, 17, 18];actual generated prefix=[16, 17, 18, 103, 361, 1306, 4742]

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

• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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, 6, 4742, 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.