Question 22507: 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=-1 and initial_values=[6, 7], 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=2, coefficients=[1, 2] in oldest-to-newest order, constant=-1 and initial_values=[6, 7], 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=2;oldest-to-newest coefficients=[1, 2];constant=-1;ALL initial a_0..a_(r-1)=[6, 7]

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, 6)

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 = 5

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(2, 7)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 14

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(5, 14)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 19

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[6, 7];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 6, 6], [1, 1, 2, 7, 14]]

new a_2=19;next state=[7, 19];drop ONLY oldest state entry then append new scalar;prefix now=[6, 7, 19]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(1, 7)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 7

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(-1, 7)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 6

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(2, 19)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 38

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(6, 38)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 44

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[7, 19];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 7, 7], [1, 2, 2, 19, 38]]

new a_3=44;next state=[19, 44];drop ONLY oldest state entry then append new scalar;prefix now=[6, 7, 19, 44]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(1, 19)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 19

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(-1, 19)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 18

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(2, 44)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 88

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(18, 88)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 106

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[19, 44];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 19, 19], [1, 3, 2, 44, 88]]

new a_4=106;next state=[44, 106];drop ONLY oldest state entry then append new scalar;prefix now=[6, 7, 19, 44, 106]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(1, 44)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 44

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(-1, 44)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 43

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(2, 106)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 212

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(43, 212)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 255

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[44, 106];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 44, 44], [1, 4, 2, 106, 212]]

new a_5=255;next state=[106, 255];drop ONLY oldest state entry then append new scalar;prefix now=[6, 7, 19, 44, 106, 255]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 := mul(1, 106)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 = 106

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(-1, 106)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 105

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 := mul(2, 255)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 = 510

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(105, 510)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 615

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[106, 255];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 106, 106], [1, 5, 2, 255, 510]]

new a_6=615;next state=[255, 615];drop ONLY oldest state entry then append new scalar;prefix now=[6, 7, 19, 44, 106, 255, 615]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 := mul(1, 255)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 = 255

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(-1, 255)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 254

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 := mul(2, 615)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 = 1230

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(254, 1230)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 1484

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[255, 615];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 1, 255, 255], [1, 6, 2, 615, 1230]]

new a_7=1484;next state=[615, 1484];drop ONLY oldest state entry then append new scalar;prefix now=[6, 7, 19, 44, 106, 255, 615, 1484]

recurrence ORIGINAL:complete ORIGINAL initial state=[6, 7];actual generated prefix=[6, 7, 19, 44, 106, 255, 615, 1484]

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, [6, 7], 7, [6, 7, 19, 44, 106, 255, 615, 1484], 1484, 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.