Question 22504: Shift the Full State and Append Each Exact New Scalar

Original question and givens

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

• G001 — Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=1 and initial_values=[3, 4], 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=8. 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=2;oldest-to-newest coefficients=[2, 4];constant=1;ALL initial a_0..a_(r-1)=[3, 4]

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, 3)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 6

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(1, 6)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 7

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

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 16

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(7, 16)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 23

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[3, 4];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 3, 6], [1, 1, 4, 4, 16]]

new a_2=23;next state=[4, 23];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(2, 4)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 8

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(1, 8)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 9

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(4, 23)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 92

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(9, 92)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 101

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[4, 23];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 4, 8], [1, 2, 4, 23, 92]]

new a_3=101;next state=[23, 101];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(2, 23)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 46

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(1, 46)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 47

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(4, 101)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 404

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(47, 404)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 451

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[23, 101];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 23, 46], [1, 3, 4, 101, 404]]

new a_4=451;next state=[101, 451];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101, 451]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(2, 101)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 202

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(1, 202)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 203

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(4, 451)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 1804

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(203, 1804)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 2007

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[101, 451];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 101, 202], [1, 4, 4, 451, 1804]]

new a_5=2007;next state=[451, 2007];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101, 451, 2007]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 := mul(2, 451)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 = 902

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(1, 902)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 903

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 := mul(4, 2007)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 = 8028

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(903, 8028)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 8931

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[451, 2007];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 451, 902], [1, 5, 4, 2007, 8028]]

new a_6=8931;next state=[2007, 8931];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101, 451, 2007, 8931]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 := mul(2, 2007)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 = 4014

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(1, 4014)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 4015

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 := mul(4, 8931)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 = 35724

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(4015, 35724)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 39739

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[2007, 8931];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 2007, 4014], [1, 6, 4, 8931, 35724]]

new a_7=39739;next state=[8931, 39739];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101, 451, 2007, 8931, 39739]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_6 := mul(2, 8931)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_6 = 17862

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(1, 17862)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 17863

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_7 := mul(4, 39739)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_7 = 158956

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(17863, 158956)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 176819

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[8931, 39739];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 2, 8931, 17862], [1, 7, 4, 39739, 158956]]

new a_8=176819;next state=[39739, 176819];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101, 451, 2007, 8931, 39739, 176819]

recurrence ORIGINAL:complete ORIGINAL initial state=[3, 4];actual generated prefix=[3, 4, 23, 101, 451, 2007, 8931, 39739, 176819]

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

• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_7 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_6 with exact rational arithmetic, not rounded decimals.
• 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, [3, 4], 8, [3, 4, 23, 101, 451, 2007, 8931, 39739, 176819], 176819, 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.