Question 22544: A Complete Initial State Gives a Unique Forward Prefix

Original question and givens

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=1 and initial_values=[18, 19], 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=2, coefficients=[1, 2] in oldest-to-newest order, constant=1 and initial_values=[18, 19], 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=2;oldest-to-newest coefficients=[1, 2];constant=1;ALL initial a_0..a_(r-1)=[18, 19]

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(1, 18)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 18

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(1, 18)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 19

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(2, 19)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 38

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(19, 38)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 57

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[18, 19];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 18, 18], [1, 1, 2, 19, 38]]

new a_2=57;next state=[19, 57];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(1, 19)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 19

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(1, 19)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 20

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(2, 57)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 114

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(20, 114)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 134

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[19, 57];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 19, 19], [1, 2, 2, 57, 114]]

new a_3=134;next state=[57, 134];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57, 134]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(1, 57)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 57

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(1, 57)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 58

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(2, 134)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 268

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(58, 268)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 326

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[57, 134];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 57, 57], [1, 3, 2, 134, 268]]

new a_4=326;next state=[134, 326];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57, 134, 326]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(1, 134)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 134

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(1, 134)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 135

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(2, 326)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 652

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(135, 652)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 787

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[134, 326];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 134, 134], [1, 4, 2, 326, 652]]

new a_5=787;next state=[326, 787];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57, 134, 326, 787]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 := mul(1, 326)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 = 326

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(1, 326)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 327

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 := mul(2, 787)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 = 1574

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(327, 1574)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 1901

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[326, 787];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 326, 326], [1, 5, 2, 787, 1574]]

new a_6=1901;next state=[787, 1901];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57, 134, 326, 787, 1901]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 := mul(1, 787)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 = 787

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(1, 787)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 788

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 := mul(2, 1901)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 = 3802

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(788, 3802)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 4590

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[787, 1901];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 1, 787, 787], [1, 6, 2, 1901, 3802]]

new a_7=4590;next state=[1901, 4590];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57, 134, 326, 787, 1901, 4590]

recurrence ORIGINAL:complete ORIGINAL initial state=[18, 19];actual generated prefix=[18, 19, 57, 134, 326, 787, 1901, 4590]

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

• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 with exact rational arithmetic, not rounded decimals.
• 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, 7, 4590, 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.