Question 22558: Different Missing-Initial Completions Give Different Whole Prefixes

Original question and givens

Use order2 coefficients=[2, 4] oldest-to-newest, constant=3, known a_0=7, alternative a_1=8 or 9, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

• G001 — Use order2 coefficients=[2, 4] oldest-to-newest, constant=3, known a_0=7, alternative a_1=8 or 9, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

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-missing-initial-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary]

recurrence LEFT completion:ORIGINAL order=2;oldest-to-newest coefficients=[2, 4];constant=3;ALL initial a_0..a_(r-1)=[7, 8]

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 LEFT completion n=2 ORIGINAL coefficient0 times a_0 := mul(2, 7)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 14

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(3, 14)

recurrence LEFT completion n=2 partial sum after coefficient0 = 17

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(4, 8)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 32

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(17, 32)

recurrence LEFT completion n=2 partial sum after coefficient1 = 49

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[7, 8];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 7, 14], [1, 1, 4, 8, 32]]

new a_2=49;next state=[8, 49];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 49]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 8)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 16

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(3, 16)

recurrence LEFT completion n=3 partial sum after coefficient0 = 19

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 49)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 196

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(19, 196)

recurrence LEFT completion n=3 partial sum after coefficient1 = 215

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[8, 49];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 8, 16], [1, 2, 4, 49, 196]]

new a_3=215;next state=[49, 215];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 49, 215]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 49)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 98

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(3, 98)

recurrence LEFT completion n=4 partial sum after coefficient0 = 101

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 215)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 860

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(101, 860)

recurrence LEFT completion n=4 partial sum after coefficient1 = 961

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[49, 215];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 49, 98], [1, 3, 4, 215, 860]]

new a_4=961;next state=[215, 961];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 49, 215, 961]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 215)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 430

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(3, 430)

recurrence LEFT completion n=5 partial sum after coefficient0 = 433

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 961)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 3844

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(433, 3844)

recurrence LEFT completion n=5 partial sum after coefficient1 = 4277

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[215, 961];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 215, 430], [1, 4, 4, 961, 3844]]

new a_5=4277;next state=[961, 4277];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 49, 215, 961, 4277]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 961)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 1922

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(3, 1922)

recurrence LEFT completion n=6 partial sum after coefficient0 = 1925

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 4277)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 17108

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(1925, 17108)

recurrence LEFT completion n=6 partial sum after coefficient1 = 19033

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[961, 4277];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 961, 1922], [1, 5, 4, 4277, 17108]]

new a_6=19033;next state=[4277, 19033];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 49, 215, 961, 4277, 19033]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 4277)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 8554

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(3, 8554)

recurrence LEFT completion n=7 partial sum after coefficient0 = 8557

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(4, 19033)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 76132

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(8557, 76132)

recurrence LEFT completion n=7 partial sum after coefficient1 = 84689

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[4277, 19033];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 4277, 8554], [1, 6, 4, 19033, 76132]]

new a_7=84689;next state=[19033, 84689];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 49, 215, 961, 4277, 19033, 84689]

recurrence LEFT completion:complete ORIGINAL initial state=[7, 8];actual generated prefix=[7, 8, 49, 215, 961, 4277, 19033, 84689]

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

recurrence RIGHT completion:ORIGINAL order=2;oldest-to-newest coefficients=[2, 4];constant=3;ALL initial a_0..a_(r-1)=[7, 9]

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 RIGHT completion n=2 ORIGINAL coefficient0 times a_0 := mul(2, 7)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 14

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(3, 14)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 17

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

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 36

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(17, 36)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 53

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[7, 9];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 7, 14], [1, 1, 4, 9, 36]]

new a_2=53;next state=[9, 53];drop ONLY oldest state entry then append new scalar;prefix now=[7, 9, 53]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 9)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 18

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(3, 18)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 21

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 53)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 212

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(21, 212)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 233

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[9, 53];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 9, 18], [1, 2, 4, 53, 212]]

new a_3=233;next state=[53, 233];drop ONLY oldest state entry then append new scalar;prefix now=[7, 9, 53, 233]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 53)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 106

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(3, 106)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 109

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 233)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 932

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(109, 932)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 1041

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[53, 233];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 53, 106], [1, 3, 4, 233, 932]]

new a_4=1041;next state=[233, 1041];drop ONLY oldest state entry then append new scalar;prefix now=[7, 9, 53, 233, 1041]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 233)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 466

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(3, 466)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 469

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 1041)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 4164

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(469, 4164)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 4633

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[233, 1041];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 233, 466], [1, 4, 4, 1041, 4164]]

new a_5=4633;next state=[1041, 4633];drop ONLY oldest state entry then append new scalar;prefix now=[7, 9, 53, 233, 1041, 4633]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 1041)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 2082

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(3, 2082)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 2085

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 4633)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 18532

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(2085, 18532)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 20617

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1041, 4633];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 1041, 2082], [1, 5, 4, 4633, 18532]]

new a_6=20617;next state=[4633, 20617];drop ONLY oldest state entry then append new scalar;prefix now=[7, 9, 53, 233, 1041, 4633, 20617]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 4633)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 9266

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(3, 9266)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 9269

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(4, 20617)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 82468

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(9269, 82468)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 91737

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[4633, 20617];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 4633, 9266], [1, 6, 4, 20617, 82468]]

new a_7=91737;next state=[20617, 91737];drop ONLY oldest state entry then append new scalar;prefix now=[7, 9, 53, 233, 1041, 4633, 20617, 91737]

recurrence RIGHT completion:complete ORIGINAL initial state=[7, 9];actual generated prefix=[7, 9, 53, 233, 1041, 4633, 20617, 91737]

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

ORIGINAL only known a_0=7;different completions a_1=8 and9;target=7

WHOLE prefixes including initials differ=Yes;actual targets=[84689, 91737] MAY coincide;one_initial_uniqueness_claim fixed No

Reverse checks

• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S063 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Recompute recurrence RIGHT completion n=6 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute recurrence RIGHT completion n=6 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Recompute recurrence RIGHT completion n=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute recurrence RIGHT completion n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S048 · premise_and_result_audit — Recompute recurrence RIGHT completion n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute recurrence RIGHT completion n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Recompute recurrence RIGHT completion n=3 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute recurrence RIGHT completion n=3 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute recurrence RIGHT completion n=2 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute recurrence RIGHT completion n=2 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 LEFT completion n=7 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute recurrence LEFT completion n=7 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute recurrence LEFT completion 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 LEFT completion n=6 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute recurrence LEFT completion n=6 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute recurrence LEFT completion 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 LEFT completion n=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute recurrence LEFT completion n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute recurrence LEFT completion 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 LEFT completion n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute recurrence LEFT completion n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute recurrence LEFT completion 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 LEFT completion n=3 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute recurrence LEFT completion n=3 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute recurrence LEFT completion 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 LEFT completion n=2 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute recurrence LEFT completion n=2 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute recurrence LEFT completion 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

(7, 8, 9, [7, 8, 49, 215, 961, 4277, 19033, 84689], [7, 9, 53, 233, 1041, 4633, 20617, 91737], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)

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.