Question 22530: 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=2 and initial_values=[4, 5, 6], 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=2 and initial_values=[4, 5, 6], 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=2;ALL initial a_0..a_(r-1)=[4, 5, 6]

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, 4)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 = 4

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(2, 4)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 6

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 := mul(2, 5)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 = 10

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(6, 10)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 16

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

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 = 18

recurrence ORIGINAL n=3 partial sum after coefficient2 := sum(16, 18)

recurrence ORIGINAL n=3 partial sum after coefficient2 = 34

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[4, 5, 6];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 4, 4], [1, 1, 2, 5, 10], [2, 2, 3, 6, 18]]

new a_3=34;next state=[5, 6, 34];drop ONLY oldest state entry then append new scalar;prefix now=[4, 5, 6, 34]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 := mul(1, 5)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 = 5

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(2, 5)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 7

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 := mul(2, 6)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 = 12

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(7, 12)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 19

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 := mul(3, 34)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 = 102

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(19, 102)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 121

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[5, 6, 34];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 5, 5], [1, 2, 2, 6, 12], [2, 3, 3, 34, 102]]

new a_4=121;next state=[6, 34, 121];drop ONLY oldest state entry then append new scalar;prefix now=[4, 5, 6, 34, 121]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 := mul(1, 6)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 = 6

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(2, 6)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 8

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 := mul(2, 34)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 = 68

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(8, 68)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 76

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 := mul(3, 121)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 = 363

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(76, 363)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 439

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[6, 34, 121];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 6, 6], [1, 3, 2, 34, 68], [2, 4, 3, 121, 363]]

new a_5=439;next state=[34, 121, 439];drop ONLY oldest state entry then append new scalar;prefix now=[4, 5, 6, 34, 121, 439]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 := mul(1, 34)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 = 34

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(2, 34)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 36

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 := mul(2, 121)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 = 242

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(36, 242)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 278

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 := mul(3, 439)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 = 1317

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(278, 1317)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 1595

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[34, 121, 439];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 34, 34], [1, 4, 2, 121, 242], [2, 5, 3, 439, 1317]]

new a_6=1595;next state=[121, 439, 1595];drop ONLY oldest state entry then append new scalar;prefix now=[4, 5, 6, 34, 121, 439, 1595]

recurrence ORIGINAL:complete ORIGINAL initial state=[4, 5, 6];actual generated prefix=[4, 5, 6, 34, 121, 439, 1595]

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, 1595, 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.