Question 22552: Different Missing-Initial Completions Give Different Whole Prefixes

Original question and givens

Use order2 coefficients=[2, 3] oldest-to-newest, constant=1, known a_0=1, alternative a_1=2 or 3, 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, 3] oldest-to-newest, constant=1, known a_0=1, alternative a_1=2 or 3, 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, 3];constant=1;ALL initial a_0..a_(r-1)=[1, 2]

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, 1)

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

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(1, 2)

recurrence LEFT completion n=2 partial sum after coefficient0 = 3

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 2)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 6

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(3, 6)

recurrence LEFT completion n=2 partial sum after coefficient1 = 9

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

new a_2=9;next state=[2, 9];drop ONLY oldest state entry then append new scalar;prefix now=[1, 2, 9]

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

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 4

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(1, 4)

recurrence LEFT completion n=3 partial sum after coefficient0 = 5

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 9)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 27

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(5, 27)

recurrence LEFT completion n=3 partial sum after coefficient1 = 32

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[2, 9];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 2, 4], [1, 2, 3, 9, 27]]

new a_3=32;next state=[9, 32];drop ONLY oldest state entry then append new scalar;prefix now=[1, 2, 9, 32]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 9)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 18

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(1, 18)

recurrence LEFT completion n=4 partial sum after coefficient0 = 19

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 32)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 96

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(19, 96)

recurrence LEFT completion n=4 partial sum after coefficient1 = 115

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[9, 32];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 9, 18], [1, 3, 3, 32, 96]]

new a_4=115;next state=[32, 115];drop ONLY oldest state entry then append new scalar;prefix now=[1, 2, 9, 32, 115]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 32)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 64

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(1, 64)

recurrence LEFT completion n=5 partial sum after coefficient0 = 65

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 115)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 345

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(65, 345)

recurrence LEFT completion n=5 partial sum after coefficient1 = 410

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[32, 115];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 32, 64], [1, 4, 3, 115, 345]]

new a_5=410;next state=[115, 410];drop ONLY oldest state entry then append new scalar;prefix now=[1, 2, 9, 32, 115, 410]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 115)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 230

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(1, 230)

recurrence LEFT completion n=6 partial sum after coefficient0 = 231

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 410)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 1230

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(231, 1230)

recurrence LEFT completion n=6 partial sum after coefficient1 = 1461

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[115, 410];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 115, 230], [1, 5, 3, 410, 1230]]

new a_6=1461;next state=[410, 1461];drop ONLY oldest state entry then append new scalar;prefix now=[1, 2, 9, 32, 115, 410, 1461]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 410)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 820

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(1, 820)

recurrence LEFT completion n=7 partial sum after coefficient0 = 821

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(3, 1461)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 4383

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(821, 4383)

recurrence LEFT completion n=7 partial sum after coefficient1 = 5204

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[410, 1461];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 410, 820], [1, 6, 3, 1461, 4383]]

new a_7=5204;next state=[1461, 5204];drop ONLY oldest state entry then append new scalar;prefix now=[1, 2, 9, 32, 115, 410, 1461, 5204]

recurrence LEFT completion:complete ORIGINAL initial state=[1, 2];actual generated prefix=[1, 2, 9, 32, 115, 410, 1461, 5204]

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, 3];constant=1;ALL initial a_0..a_(r-1)=[1, 3]

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, 1)

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

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(1, 2)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 3

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 3)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 9

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(3, 9)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 12

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

new a_2=12;next state=[3, 12];drop ONLY oldest state entry then append new scalar;prefix now=[1, 3, 12]

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

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 6

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(1, 6)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 7

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 12)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 36

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(7, 36)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 43

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[3, 12];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 3, 6], [1, 2, 3, 12, 36]]

new a_3=43;next state=[12, 43];drop ONLY oldest state entry then append new scalar;prefix now=[1, 3, 12, 43]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 12)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 24

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(1, 24)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 25

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 43)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 129

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(25, 129)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 154

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[12, 43];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 12, 24], [1, 3, 3, 43, 129]]

new a_4=154;next state=[43, 154];drop ONLY oldest state entry then append new scalar;prefix now=[1, 3, 12, 43, 154]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 43)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 86

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(1, 86)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 87

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 154)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 462

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(87, 462)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 549

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[43, 154];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 43, 86], [1, 4, 3, 154, 462]]

new a_5=549;next state=[154, 549];drop ONLY oldest state entry then append new scalar;prefix now=[1, 3, 12, 43, 154, 549]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 154)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 308

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(1, 308)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 309

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 549)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 1647

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(309, 1647)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 1956

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[154, 549];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 154, 308], [1, 5, 3, 549, 1647]]

new a_6=1956;next state=[549, 1956];drop ONLY oldest state entry then append new scalar;prefix now=[1, 3, 12, 43, 154, 549, 1956]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 549)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 1098

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(1, 1098)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 1099

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(3, 1956)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 5868

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(1099, 5868)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 6967

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[549, 1956];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 549, 1098], [1, 6, 3, 1956, 5868]]

new a_7=6967;next state=[1956, 6967];drop ONLY oldest state entry then append new scalar;prefix now=[1, 3, 12, 43, 154, 549, 1956, 6967]

recurrence RIGHT completion:complete ORIGINAL initial state=[1, 3];actual generated prefix=[1, 3, 12, 43, 154, 549, 1956, 6967]

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=1;different completions a_1=2 and3;target=7

WHOLE prefixes including initials differ=Yes;actual targets=[5204, 6967] 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

(1, 2, 3, [1, 2, 9, 32, 115, 410, 1461, 5204], [1, 3, 12, 43, 154, 549, 1956, 6967], 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.