Question 22535: 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=2 and initial_values=[9, 10], 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=2 and initial_values=[9, 10], 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=2;ALL initial a_0..a_(r-1)=[9, 10]

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, 9)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 18

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(2, 18)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 20

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(4, 10)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 40

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(20, 40)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 60

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[9, 10];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 9, 18], [1, 1, 4, 10, 40]]

new a_2=60;next state=[10, 60];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 60]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(2, 10)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 20

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(2, 20)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 22

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(4, 60)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 240

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(22, 240)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 262

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[10, 60];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 10, 20], [1, 2, 4, 60, 240]]

new a_3=262;next state=[60, 262];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 60, 262]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(2, 60)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 120

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(2, 120)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 122

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(4, 262)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 1048

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(122, 1048)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 1170

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[60, 262];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 60, 120], [1, 3, 4, 262, 1048]]

new a_4=1170;next state=[262, 1170];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 60, 262, 1170]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(2, 262)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 524

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(2, 524)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 526

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(4, 1170)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 4680

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(526, 4680)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 5206

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[262, 1170];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 262, 524], [1, 4, 4, 1170, 4680]]

new a_5=5206;next state=[1170, 5206];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 60, 262, 1170, 5206]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 := mul(2, 1170)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 = 2340

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(2, 2340)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 2342

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 := mul(4, 5206)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 = 20824

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(2342, 20824)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 23166

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1170, 5206];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 1170, 2340], [1, 5, 4, 5206, 20824]]

new a_6=23166;next state=[5206, 23166];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 60, 262, 1170, 5206, 23166]

recurrence ORIGINAL:complete ORIGINAL initial state=[9, 10];actual generated prefix=[9, 10, 60, 262, 1170, 5206, 23166]

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, 23166, 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.