Question 22568: Different Missing-Initial Completions Give Different Whole Prefixes

Original question and givens

Use order2 coefficients=[3, 4] oldest-to-newest, constant=1, known a_0=17, alternative a_1=18 or 19, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

• G001 — Use order2 coefficients=[3, 4] oldest-to-newest, constant=1, known a_0=17, alternative a_1=18 or 19, target_n=8. 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=[3, 4];constant=1;ALL initial a_0..a_(r-1)=[17, 18]

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(3, 17)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 51

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(1, 51)

recurrence LEFT completion n=2 partial sum after coefficient0 = 52

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(4, 18)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 72

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(52, 72)

recurrence LEFT completion n=2 partial sum after coefficient1 = 124

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[17, 18];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 17, 51], [1, 1, 4, 18, 72]]

new a_2=124;next state=[18, 124];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 124]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 18)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 54

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(1, 54)

recurrence LEFT completion n=3 partial sum after coefficient0 = 55

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 124)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 496

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(55, 496)

recurrence LEFT completion n=3 partial sum after coefficient1 = 551

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[18, 124];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 18, 54], [1, 2, 4, 124, 496]]

new a_3=551;next state=[124, 551];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 124, 551]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 124)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 372

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(1, 372)

recurrence LEFT completion n=4 partial sum after coefficient0 = 373

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 551)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 2204

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(373, 2204)

recurrence LEFT completion n=4 partial sum after coefficient1 = 2577

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[124, 551];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 124, 372], [1, 3, 4, 551, 2204]]

new a_4=2577;next state=[551, 2577];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 124, 551, 2577]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 551)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 1653

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(1, 1653)

recurrence LEFT completion n=5 partial sum after coefficient0 = 1654

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 2577)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 10308

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(1654, 10308)

recurrence LEFT completion n=5 partial sum after coefficient1 = 11962

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[551, 2577];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 551, 1653], [1, 4, 4, 2577, 10308]]

new a_5=11962;next state=[2577, 11962];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 124, 551, 2577, 11962]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 2577)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 7731

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(1, 7731)

recurrence LEFT completion n=6 partial sum after coefficient0 = 7732

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 11962)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 47848

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(7732, 47848)

recurrence LEFT completion n=6 partial sum after coefficient1 = 55580

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[2577, 11962];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 2577, 7731], [1, 5, 4, 11962, 47848]]

new a_6=55580;next state=[11962, 55580];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 124, 551, 2577, 11962, 55580]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 11962)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 35886

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(1, 35886)

recurrence LEFT completion n=7 partial sum after coefficient0 = 35887

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(4, 55580)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 222320

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(35887, 222320)

recurrence LEFT completion n=7 partial sum after coefficient1 = 258207

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[11962, 55580];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 11962, 35886], [1, 6, 4, 55580, 222320]]

new a_7=258207;next state=[55580, 258207];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 124, 551, 2577, 11962, 55580, 258207]

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 55580)

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 = 166740

recurrence LEFT completion n=8 partial sum after coefficient0 := sum(1, 166740)

recurrence LEFT completion n=8 partial sum after coefficient0 = 166741

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 := mul(4, 258207)

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 = 1032828

recurrence LEFT completion n=8 partial sum after coefficient1 := sum(166741, 1032828)

recurrence LEFT completion n=8 partial sum after coefficient1 = 1199569

recurrence LEFT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[55580, 258207];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 55580, 166740], [1, 7, 4, 258207, 1032828]]

new a_8=1199569;next state=[258207, 1199569];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 124, 551, 2577, 11962, 55580, 258207, 1199569]

recurrence LEFT completion:complete ORIGINAL initial state=[17, 18];actual generated prefix=[17, 18, 124, 551, 2577, 11962, 55580, 258207, 1199569]

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=[3, 4];constant=1;ALL initial a_0..a_(r-1)=[17, 19]

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(3, 17)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 51

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(1, 51)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 52

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(4, 19)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 76

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(52, 76)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 128

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[17, 19];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 17, 51], [1, 1, 4, 19, 76]]

new a_2=128;next state=[19, 128];drop ONLY oldest state entry then append new scalar;prefix now=[17, 19, 128]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 19)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 57

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(1, 57)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 58

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 128)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 512

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(58, 512)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 570

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[19, 128];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 19, 57], [1, 2, 4, 128, 512]]

new a_3=570;next state=[128, 570];drop ONLY oldest state entry then append new scalar;prefix now=[17, 19, 128, 570]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 128)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 384

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(1, 384)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 385

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 570)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 2280

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(385, 2280)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 2665

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[128, 570];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 128, 384], [1, 3, 4, 570, 2280]]

new a_4=2665;next state=[570, 2665];drop ONLY oldest state entry then append new scalar;prefix now=[17, 19, 128, 570, 2665]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 570)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 1710

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(1, 1710)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 1711

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 2665)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 10660

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(1711, 10660)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 12371

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[570, 2665];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 570, 1710], [1, 4, 4, 2665, 10660]]

new a_5=12371;next state=[2665, 12371];drop ONLY oldest state entry then append new scalar;prefix now=[17, 19, 128, 570, 2665, 12371]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 2665)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 7995

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(1, 7995)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 7996

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 12371)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 49484

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(7996, 49484)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 57480

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[2665, 12371];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 2665, 7995], [1, 5, 4, 12371, 49484]]

new a_6=57480;next state=[12371, 57480];drop ONLY oldest state entry then append new scalar;prefix now=[17, 19, 128, 570, 2665, 12371, 57480]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 12371)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 37113

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(1, 37113)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 37114

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(4, 57480)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 229920

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(37114, 229920)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 267034

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[12371, 57480];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 12371, 37113], [1, 6, 4, 57480, 229920]]

new a_7=267034;next state=[57480, 267034];drop ONLY oldest state entry then append new scalar;prefix now=[17, 19, 128, 570, 2665, 12371, 57480, 267034]

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 57480)

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 = 172440

recurrence RIGHT completion n=8 partial sum after coefficient0 := sum(1, 172440)

recurrence RIGHT completion n=8 partial sum after coefficient0 = 172441

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 := mul(4, 267034)

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 = 1068136

recurrence RIGHT completion n=8 partial sum after coefficient1 := sum(172441, 1068136)

recurrence RIGHT completion n=8 partial sum after coefficient1 = 1240577

recurrence RIGHT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[57480, 267034];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 57480, 172440], [1, 7, 4, 267034, 1068136]]

new a_8=1240577;next state=[267034, 1240577];drop ONLY oldest state entry then append new scalar;prefix now=[17, 19, 128, 570, 2665, 12371, 57480, 267034, 1240577]

recurrence RIGHT completion:complete ORIGINAL initial state=[17, 19];actual generated prefix=[17, 19, 128, 570, 2665, 12371, 57480, 267034, 1240577]

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=17;different completions a_1=18 and19;target=8

WHOLE prefixes including initials differ=Yes;actual targets=[1199569, 1240577] MAY coincide;one_initial_uniqueness_claim fixed No

Reverse checks

• S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S073 · premise_and_result_audit — Recompute recurrence RIGHT completion n=8 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute recurrence RIGHT completion n=8 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S068 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 with exact rational arithmetic, not rounded decimals.
• 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=6 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute recurrence RIGHT completion n=6 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 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=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute recurrence RIGHT completion n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 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=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute recurrence RIGHT completion n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 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=3 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute recurrence RIGHT completion n=3 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 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=2 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute recurrence RIGHT completion n=2 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 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 — 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 LEFT completion n=8 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute recurrence LEFT completion n=8 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute recurrence LEFT completion 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 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

(17, 18, 19, [17, 18, 124, 551, 2577, 11962, 55580, 258207, 1199569], [17, 19, 128, 570, 2665, 12371, 57480, 267034, 1240577], 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.