Question 22556: Different Missing-Initial Completions Give Different Whole Prefixes

Original question and givens

Use order2 coefficients=[3, 2] oldest-to-newest, constant=1, known a_0=5, alternative a_1=6 or 7, 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, 2] oldest-to-newest, constant=1, known a_0=5, alternative a_1=6 or 7, 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, 2];constant=1;ALL initial a_0..a_(r-1)=[5, 6]

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, 5)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 15

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(1, 15)

recurrence LEFT completion n=2 partial sum after coefficient0 = 16

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(2, 6)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 12

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(16, 12)

recurrence LEFT completion n=2 partial sum after coefficient1 = 28

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[5, 6];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 5, 15], [1, 1, 2, 6, 12]]

new a_2=28;next state=[6, 28];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 28]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 6)

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

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(1, 18)

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

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(2, 28)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 56

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(19, 56)

recurrence LEFT completion n=3 partial sum after coefficient1 = 75

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[6, 28];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 6, 18], [1, 2, 2, 28, 56]]

new a_3=75;next state=[28, 75];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 28, 75]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 28)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 84

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(1, 84)

recurrence LEFT completion n=4 partial sum after coefficient0 = 85

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(2, 75)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 150

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(85, 150)

recurrence LEFT completion n=4 partial sum after coefficient1 = 235

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[28, 75];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 28, 84], [1, 3, 2, 75, 150]]

new a_4=235;next state=[75, 235];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 28, 75, 235]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 75)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 225

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(1, 225)

recurrence LEFT completion n=5 partial sum after coefficient0 = 226

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(2, 235)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 470

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(226, 470)

recurrence LEFT completion n=5 partial sum after coefficient1 = 696

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[75, 235];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 75, 225], [1, 4, 2, 235, 470]]

new a_5=696;next state=[235, 696];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 28, 75, 235, 696]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 235)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 705

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(1, 705)

recurrence LEFT completion n=6 partial sum after coefficient0 = 706

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(2, 696)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 1392

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(706, 1392)

recurrence LEFT completion n=6 partial sum after coefficient1 = 2098

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[235, 696];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 235, 705], [1, 5, 2, 696, 1392]]

new a_6=2098;next state=[696, 2098];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 28, 75, 235, 696, 2098]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 696)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 2088

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(1, 2088)

recurrence LEFT completion n=7 partial sum after coefficient0 = 2089

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(2, 2098)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 4196

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(2089, 4196)

recurrence LEFT completion n=7 partial sum after coefficient1 = 6285

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[696, 2098];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 696, 2088], [1, 6, 2, 2098, 4196]]

new a_7=6285;next state=[2098, 6285];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 28, 75, 235, 696, 2098, 6285]

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 2098)

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 = 6294

recurrence LEFT completion n=8 partial sum after coefficient0 := sum(1, 6294)

recurrence LEFT completion n=8 partial sum after coefficient0 = 6295

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 := mul(2, 6285)

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 = 12570

recurrence LEFT completion n=8 partial sum after coefficient1 := sum(6295, 12570)

recurrence LEFT completion n=8 partial sum after coefficient1 = 18865

recurrence LEFT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[2098, 6285];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 2098, 6294], [1, 7, 2, 6285, 12570]]

new a_8=18865;next state=[6285, 18865];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 28, 75, 235, 696, 2098, 6285, 18865]

recurrence LEFT completion:complete ORIGINAL initial state=[5, 6];actual generated prefix=[5, 6, 28, 75, 235, 696, 2098, 6285, 18865]

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, 2];constant=1;ALL initial a_0..a_(r-1)=[5, 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 RIGHT completion n=2 ORIGINAL coefficient0 times a_0 := mul(3, 5)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 15

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(1, 15)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 16

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

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 14

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(16, 14)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 30

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[5, 7];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 5, 15], [1, 1, 2, 7, 14]]

new a_2=30;next state=[7, 30];drop ONLY oldest state entry then append new scalar;prefix now=[5, 7, 30]

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

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 21

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(1, 21)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 22

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(2, 30)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 60

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(22, 60)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 82

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[7, 30];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 7, 21], [1, 2, 2, 30, 60]]

new a_3=82;next state=[30, 82];drop ONLY oldest state entry then append new scalar;prefix now=[5, 7, 30, 82]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 30)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 90

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(1, 90)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 91

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(2, 82)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 164

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(91, 164)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 255

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[30, 82];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 30, 90], [1, 3, 2, 82, 164]]

new a_4=255;next state=[82, 255];drop ONLY oldest state entry then append new scalar;prefix now=[5, 7, 30, 82, 255]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 82)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 246

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(1, 246)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 247

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(2, 255)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 510

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(247, 510)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 757

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[82, 255];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 82, 246], [1, 4, 2, 255, 510]]

new a_5=757;next state=[255, 757];drop ONLY oldest state entry then append new scalar;prefix now=[5, 7, 30, 82, 255, 757]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 255)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 765

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(1, 765)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 766

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(2, 757)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 1514

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(766, 1514)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 2280

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[255, 757];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 255, 765], [1, 5, 2, 757, 1514]]

new a_6=2280;next state=[757, 2280];drop ONLY oldest state entry then append new scalar;prefix now=[5, 7, 30, 82, 255, 757, 2280]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 757)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 2271

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(1, 2271)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 2272

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(2, 2280)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 4560

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(2272, 4560)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 6832

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[757, 2280];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 757, 2271], [1, 6, 2, 2280, 4560]]

new a_7=6832;next state=[2280, 6832];drop ONLY oldest state entry then append new scalar;prefix now=[5, 7, 30, 82, 255, 757, 2280, 6832]

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 2280)

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 = 6840

recurrence RIGHT completion n=8 partial sum after coefficient0 := sum(1, 6840)

recurrence RIGHT completion n=8 partial sum after coefficient0 = 6841

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 := mul(2, 6832)

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 = 13664

recurrence RIGHT completion n=8 partial sum after coefficient1 := sum(6841, 13664)

recurrence RIGHT completion n=8 partial sum after coefficient1 = 20505

recurrence RIGHT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[2280, 6832];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 2280, 6840], [1, 7, 2, 6832, 13664]]

new a_8=20505;next state=[6832, 20505];drop ONLY oldest state entry then append new scalar;prefix now=[5, 7, 30, 82, 255, 757, 2280, 6832, 20505]

recurrence RIGHT completion:complete ORIGINAL initial state=[5, 7];actual generated prefix=[5, 7, 30, 82, 255, 757, 2280, 6832, 20505]

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=5;different completions a_1=6 and7;target=8

WHOLE prefixes including initials differ=Yes;actual targets=[18865, 20505] 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

(5, 6, 7, [5, 6, 28, 75, 235, 696, 2098, 6285, 18865], [5, 7, 30, 82, 255, 757, 2280, 6832, 20505], 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.