Question 22516: 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=-2 and initial_values=[15, 16], 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=-2 and initial_values=[15, 16], 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=-2;ALL initial a_0..a_(r-1)=[15, 16]

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, 15)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 30

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(-2, 30)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 28

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(4, 16)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 64

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(28, 64)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 92

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[15, 16];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 15, 30], [1, 1, 4, 16, 64]]

new a_2=92;next state=[16, 92];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 92]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(2, 16)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 32

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(-2, 32)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 30

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(4, 92)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 368

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(30, 368)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 398

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[16, 92];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 16, 32], [1, 2, 4, 92, 368]]

new a_3=398;next state=[92, 398];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 92, 398]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(2, 92)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 184

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(-2, 184)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 182

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(4, 398)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 1592

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(182, 1592)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 1774

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[92, 398];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 92, 184], [1, 3, 4, 398, 1592]]

new a_4=1774;next state=[398, 1774];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 92, 398, 1774]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(2, 398)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 796

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(-2, 796)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 794

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(4, 1774)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 7096

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(794, 7096)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 7890

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[398, 1774];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 398, 796], [1, 4, 4, 1774, 7096]]

new a_5=7890;next state=[1774, 7890];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 92, 398, 1774, 7890]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 := mul(2, 1774)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 = 3548

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(-2, 3548)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 3546

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 := mul(4, 7890)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 = 31560

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(3546, 31560)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 35106

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1774, 7890];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 1774, 3548], [1, 5, 4, 7890, 31560]]

new a_6=35106;next state=[7890, 35106];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 92, 398, 1774, 7890, 35106]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 := mul(2, 7890)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 = 15780

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(-2, 15780)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 15778

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 := mul(4, 35106)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 = 140424

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(15778, 140424)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 156202

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[7890, 35106];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 7890, 15780], [1, 6, 4, 35106, 140424]]

new a_7=156202;next state=[35106, 156202];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 92, 398, 1774, 7890, 35106, 156202]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_6 := mul(2, 35106)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_6 = 70212

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(-2, 70212)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 70210

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_7 := mul(4, 156202)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_7 = 624808

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(70210, 624808)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 695018

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[35106, 156202];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 2, 35106, 70212], [1, 7, 4, 156202, 624808]]

new a_8=695018;next state=[156202, 695018];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 92, 398, 1774, 7890, 35106, 156202, 695018]

recurrence ORIGINAL:complete ORIGINAL initial state=[15, 16];actual generated prefix=[15, 16, 92, 398, 1774, 7890, 35106, 156202, 695018]

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, [15, 16], 8, [15, 16, 92, 398, 1774, 7890, 35106, 156202, 695018], 695018, 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.