Question 22565: Different Missing-Initial Completions Give Different Whole Prefixes

Original question and givens

Use order2 coefficients=[3, 6] oldest-to-newest, constant=2, known a_0=14, alternative a_1=15 or 16, 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, 6] oldest-to-newest, constant=2, known a_0=14, alternative a_1=15 or 16, 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, 6];constant=2;ALL initial a_0..a_(r-1)=[14, 15]

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, 14)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 42

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(2, 42)

recurrence LEFT completion n=2 partial sum after coefficient0 = 44

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(6, 15)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 90

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(44, 90)

recurrence LEFT completion n=2 partial sum after coefficient1 = 134

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[14, 15];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 14, 42], [1, 1, 6, 15, 90]]

new a_2=134;next state=[15, 134];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 134]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 15)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 45

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(2, 45)

recurrence LEFT completion n=3 partial sum after coefficient0 = 47

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(6, 134)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 804

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(47, 804)

recurrence LEFT completion n=3 partial sum after coefficient1 = 851

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[15, 134];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 15, 45], [1, 2, 6, 134, 804]]

new a_3=851;next state=[134, 851];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 134, 851]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 134)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 402

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(2, 402)

recurrence LEFT completion n=4 partial sum after coefficient0 = 404

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(6, 851)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 5106

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(404, 5106)

recurrence LEFT completion n=4 partial sum after coefficient1 = 5510

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[134, 851];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 134, 402], [1, 3, 6, 851, 5106]]

new a_4=5510;next state=[851, 5510];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 134, 851, 5510]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 851)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 2553

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(2, 2553)

recurrence LEFT completion n=5 partial sum after coefficient0 = 2555

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(6, 5510)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 33060

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(2555, 33060)

recurrence LEFT completion n=5 partial sum after coefficient1 = 35615

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[851, 5510];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 851, 2553], [1, 4, 6, 5510, 33060]]

new a_5=35615;next state=[5510, 35615];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 134, 851, 5510, 35615]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 5510)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 16530

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(2, 16530)

recurrence LEFT completion n=6 partial sum after coefficient0 = 16532

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(6, 35615)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 213690

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(16532, 213690)

recurrence LEFT completion n=6 partial sum after coefficient1 = 230222

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[5510, 35615];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 5510, 16530], [1, 5, 6, 35615, 213690]]

new a_6=230222;next state=[35615, 230222];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 134, 851, 5510, 35615, 230222]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 35615)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 106845

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(2, 106845)

recurrence LEFT completion n=7 partial sum after coefficient0 = 106847

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(6, 230222)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 1381332

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(106847, 1381332)

recurrence LEFT completion n=7 partial sum after coefficient1 = 1488179

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[35615, 230222];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 35615, 106845], [1, 6, 6, 230222, 1381332]]

new a_7=1488179;next state=[230222, 1488179];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 134, 851, 5510, 35615, 230222, 1488179]

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 230222)

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 = 690666

recurrence LEFT completion n=8 partial sum after coefficient0 := sum(2, 690666)

recurrence LEFT completion n=8 partial sum after coefficient0 = 690668

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 := mul(6, 1488179)

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 = 8929074

recurrence LEFT completion n=8 partial sum after coefficient1 := sum(690668, 8929074)

recurrence LEFT completion n=8 partial sum after coefficient1 = 9619742

recurrence LEFT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[230222, 1488179];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 230222, 690666], [1, 7, 6, 1488179, 8929074]]

new a_8=9619742;next state=[1488179, 9619742];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 134, 851, 5510, 35615, 230222, 1488179, 9619742]

recurrence LEFT completion:complete ORIGINAL initial state=[14, 15];actual generated prefix=[14, 15, 134, 851, 5510, 35615, 230222, 1488179, 9619742]

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, 6];constant=2;ALL initial a_0..a_(r-1)=[14, 16]

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, 14)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 42

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(2, 42)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 44

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(6, 16)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 96

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(44, 96)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 140

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[14, 16];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 14, 42], [1, 1, 6, 16, 96]]

new a_2=140;next state=[16, 140];drop ONLY oldest state entry then append new scalar;prefix now=[14, 16, 140]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 16)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 48

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(2, 48)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 50

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(6, 140)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 840

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(50, 840)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 890

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[16, 140];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 16, 48], [1, 2, 6, 140, 840]]

new a_3=890;next state=[140, 890];drop ONLY oldest state entry then append new scalar;prefix now=[14, 16, 140, 890]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 140)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 420

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(2, 420)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 422

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(6, 890)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 5340

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(422, 5340)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 5762

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[140, 890];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 140, 420], [1, 3, 6, 890, 5340]]

new a_4=5762;next state=[890, 5762];drop ONLY oldest state entry then append new scalar;prefix now=[14, 16, 140, 890, 5762]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 890)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 2670

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(2, 2670)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 2672

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(6, 5762)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 34572

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(2672, 34572)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 37244

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[890, 5762];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 890, 2670], [1, 4, 6, 5762, 34572]]

new a_5=37244;next state=[5762, 37244];drop ONLY oldest state entry then append new scalar;prefix now=[14, 16, 140, 890, 5762, 37244]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 5762)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 17286

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(2, 17286)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 17288

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(6, 37244)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 223464

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(17288, 223464)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 240752

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[5762, 37244];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 5762, 17286], [1, 5, 6, 37244, 223464]]

new a_6=240752;next state=[37244, 240752];drop ONLY oldest state entry then append new scalar;prefix now=[14, 16, 140, 890, 5762, 37244, 240752]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 37244)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 111732

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(2, 111732)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 111734

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(6, 240752)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 1444512

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(111734, 1444512)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 1556246

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[37244, 240752];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 37244, 111732], [1, 6, 6, 240752, 1444512]]

new a_7=1556246;next state=[240752, 1556246];drop ONLY oldest state entry then append new scalar;prefix now=[14, 16, 140, 890, 5762, 37244, 240752, 1556246]

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 240752)

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 = 722256

recurrence RIGHT completion n=8 partial sum after coefficient0 := sum(2, 722256)

recurrence RIGHT completion n=8 partial sum after coefficient0 = 722258

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 := mul(6, 1556246)

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 = 9337476

recurrence RIGHT completion n=8 partial sum after coefficient1 := sum(722258, 9337476)

recurrence RIGHT completion n=8 partial sum after coefficient1 = 10059734

recurrence RIGHT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[240752, 1556246];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 240752, 722256], [1, 7, 6, 1556246, 9337476]]

new a_8=10059734;next state=[1556246, 10059734];drop ONLY oldest state entry then append new scalar;prefix now=[14, 16, 140, 890, 5762, 37244, 240752, 1556246, 10059734]

recurrence RIGHT completion:complete ORIGINAL initial state=[14, 16];actual generated prefix=[14, 16, 140, 890, 5762, 37244, 240752, 1556246, 10059734]

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=14;different completions a_1=15 and16;target=8

WHOLE prefixes including initials differ=Yes;actual targets=[9619742, 10059734] 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

(14, 15, 16, [14, 15, 134, 851, 5510, 35615, 230222, 1488179, 9619742], [14, 16, 140, 890, 5762, 37244, 240752, 1556246, 10059734], 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.