Question 22509: Shift the Full State and Append Each Exact New Scalar

Original question and givens

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=1 and initial_values=[8, 9, 10, 11], 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_values, target_n, prefix, target_value, certificate).

• G001 — Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=1 and initial_values=[8, 9, 10, 11], 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_values, target_n, prefix, target_value, 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-forward-generation;original snapshot and mathematics retained

explicit ordered source output fields=[order, initial_values, target_n, prefix, target_value, certificate]

recurrence ORIGINAL:ORIGINAL order=4;oldest-to-newest coefficients=[1, 2, 3, 4];constant=1;ALL initial a_0..a_(r-1)=[8, 9, 10, 11]

for EVERY n>=4,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=4;forward rule is total and single-valued

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 := mul(1, 8)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 8

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(1, 8)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 9

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(2, 9)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 18

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(9, 18)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 27

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(3, 10)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 30

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(27, 30)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 57

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(4, 11)

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 44

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(57, 44)

recurrence ORIGINAL n=4 partial sum after coefficient3 = 101

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[8, 9, 10, 11];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 8, 8], [1, 1, 2, 9, 18], [2, 2, 3, 10, 30], [3, 3, 4, 11, 44]]

new a_4=101;next state=[9, 10, 11, 101];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 10, 11, 101]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(1, 9)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 9

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(1, 9)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 10

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(2, 10)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 20

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(10, 20)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 30

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(3, 11)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 33

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(30, 33)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 63

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(4, 101)

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 404

recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(63, 404)

recurrence ORIGINAL n=5 partial sum after coefficient3 = 467

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[9, 10, 11, 101];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 9, 9], [1, 2, 2, 10, 20], [2, 3, 3, 11, 33], [3, 4, 4, 101, 404]]

new a_5=467;next state=[10, 11, 101, 467];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 10, 11, 101, 467]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(1, 10)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 10

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(1, 10)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 11

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(2, 11)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 22

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(11, 22)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 33

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(3, 101)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 303

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(33, 303)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 336

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(4, 467)

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 1868

recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(336, 1868)

recurrence ORIGINAL n=6 partial sum after coefficient3 = 2204

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[10, 11, 101, 467];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 10, 10], [1, 3, 2, 11, 22], [2, 4, 3, 101, 303], [3, 5, 4, 467, 1868]]

new a_6=2204;next state=[11, 101, 467, 2204];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 10, 11, 101, 467, 2204]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(1, 11)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 11

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(1, 11)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 12

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(2, 101)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 202

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(12, 202)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 214

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(3, 467)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 1401

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(214, 1401)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 1615

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(4, 2204)

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 8816

recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(1615, 8816)

recurrence ORIGINAL n=7 partial sum after coefficient3 = 10431

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[11, 101, 467, 2204];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 11, 11], [1, 4, 2, 101, 202], [2, 5, 3, 467, 1401], [3, 6, 4, 2204, 8816]]

new a_7=10431;next state=[101, 467, 2204, 10431];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 10, 11, 101, 467, 2204, 10431]

recurrence ORIGINAL:complete ORIGINAL initial state=[8, 9, 10, 11];actual generated prefix=[8, 9, 10, 11, 101, 467, 2204, 10431]

ANY two sequences with SAME r initials and SAME total rule agree at n=4 by substitution,then at every later n by induction;this determines the requested prefix only

Reverse checks

• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute recurrence ORIGINAL n=7 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 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

(4, [8, 9, 10, 11], 7, [8, 9, 10, 11, 101, 467, 2204, 10431], 10431, exact_forward_application_only)

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.