Question 22501: Shift the Full State and Append Each Exact New Scalar

Original question and givens

Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-2 and initial_values=[0, 1], 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=5. Give (order, initial_values, target_n, prefix, target_value, certificate).

• G001 — Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-2 and initial_values=[0, 1], 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=5. 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=[1, 2];constant=-2;ALL initial a_0..a_(r-1)=[0, 1]

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, 0)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 0

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(-2, 0)

recurrence ORIGINAL n=2 partial sum after coefficient0 = -2

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(2, 1)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 2

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

recurrence ORIGINAL n=2 partial sum after coefficient1 = 0

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

new a_2=0;next state=[1, 0];drop ONLY oldest state entry then append new scalar;prefix now=[0, 1, 0]

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

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 1

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(-2, 1)

recurrence ORIGINAL n=3 partial sum after coefficient0 = -1

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(2, 0)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 0

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(-1, 0)

recurrence ORIGINAL n=3 partial sum after coefficient1 = -1

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

new a_3=-1;next state=[0, -1];drop ONLY oldest state entry then append new scalar;prefix now=[0, 1, 0, -1]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(1, 0)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 0

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(-2, 0)

recurrence ORIGINAL n=4 partial sum after coefficient0 = -2

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(2, -1)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = -2

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(-2, -2)

recurrence ORIGINAL n=4 partial sum after coefficient1 = -4

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

new a_4=-4;next state=[-1, -4];drop ONLY oldest state entry then append new scalar;prefix now=[0, 1, 0, -1, -4]

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

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = -1

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(-2, -1)

recurrence ORIGINAL n=5 partial sum after coefficient0 = -3

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(2, -4)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = -8

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(-3, -8)

recurrence ORIGINAL n=5 partial sum after coefficient1 = -11

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[-1, -4];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, -1, -1], [1, 4, 2, -4, -8]]

new a_5=-11;next state=[-4, -11];drop ONLY oldest state entry then append new scalar;prefix now=[0, 1, 0, -1, -4, -11]

recurrence ORIGINAL:complete ORIGINAL initial state=[0, 1];actual generated prefix=[0, 1, 0, -1, -4, -11]

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

• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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, [0, 1], 5, [0, 1, 0, -1, -4, -11], -11, 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.