Question 22570: Different Missing-Initial Completions Give Different Whole Prefixes

Original question and givens

Use order2 coefficients=[2, 6] oldest-to-newest, constant=3, known a_0=19, alternative a_1=20 or 21, 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, 6] oldest-to-newest, constant=3, known a_0=19, alternative a_1=20 or 21, 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, 6];constant=3;ALL initial a_0..a_(r-1)=[19, 20]

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, 19)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 38

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(3, 38)

recurrence LEFT completion n=2 partial sum after coefficient0 = 41

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(6, 20)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 120

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(41, 120)

recurrence LEFT completion n=2 partial sum after coefficient1 = 161

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[19, 20];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 19, 38], [1, 1, 6, 20, 120]]

new a_2=161;next state=[20, 161];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 161]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 20)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 40

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(3, 40)

recurrence LEFT completion n=3 partial sum after coefficient0 = 43

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(6, 161)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 966

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(43, 966)

recurrence LEFT completion n=3 partial sum after coefficient1 = 1009

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[20, 161];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 20, 40], [1, 2, 6, 161, 966]]

new a_3=1009;next state=[161, 1009];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 161, 1009]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 161)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 322

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(3, 322)

recurrence LEFT completion n=4 partial sum after coefficient0 = 325

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(6, 1009)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 6054

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(325, 6054)

recurrence LEFT completion n=4 partial sum after coefficient1 = 6379

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[161, 1009];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 161, 322], [1, 3, 6, 1009, 6054]]

new a_4=6379;next state=[1009, 6379];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 161, 1009, 6379]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 1009)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 2018

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(3, 2018)

recurrence LEFT completion n=5 partial sum after coefficient0 = 2021

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(6, 6379)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 38274

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(2021, 38274)

recurrence LEFT completion n=5 partial sum after coefficient1 = 40295

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[1009, 6379];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 1009, 2018], [1, 4, 6, 6379, 38274]]

new a_5=40295;next state=[6379, 40295];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 161, 1009, 6379, 40295]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 6379)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 12758

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(3, 12758)

recurrence LEFT completion n=6 partial sum after coefficient0 = 12761

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(6, 40295)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 241770

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(12761, 241770)

recurrence LEFT completion n=6 partial sum after coefficient1 = 254531

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[6379, 40295];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 6379, 12758], [1, 5, 6, 40295, 241770]]

new a_6=254531;next state=[40295, 254531];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 161, 1009, 6379, 40295, 254531]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 40295)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 80590

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(3, 80590)

recurrence LEFT completion n=7 partial sum after coefficient0 = 80593

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(6, 254531)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 1527186

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(80593, 1527186)

recurrence LEFT completion n=7 partial sum after coefficient1 = 1607779

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[40295, 254531];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 40295, 80590], [1, 6, 6, 254531, 1527186]]

new a_7=1607779;next state=[254531, 1607779];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 161, 1009, 6379, 40295, 254531, 1607779]

recurrence LEFT completion:complete ORIGINAL initial state=[19, 20];actual generated prefix=[19, 20, 161, 1009, 6379, 40295, 254531, 1607779]

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, 6];constant=3;ALL initial a_0..a_(r-1)=[19, 21]

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, 19)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 38

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(3, 38)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 41

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(6, 21)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 126

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(41, 126)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 167

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[19, 21];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 19, 38], [1, 1, 6, 21, 126]]

new a_2=167;next state=[21, 167];drop ONLY oldest state entry then append new scalar;prefix now=[19, 21, 167]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 21)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 42

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(3, 42)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 45

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(6, 167)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 1002

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(45, 1002)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 1047

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[21, 167];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 21, 42], [1, 2, 6, 167, 1002]]

new a_3=1047;next state=[167, 1047];drop ONLY oldest state entry then append new scalar;prefix now=[19, 21, 167, 1047]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 167)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 334

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(3, 334)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 337

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(6, 1047)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 6282

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(337, 6282)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 6619

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[167, 1047];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 167, 334], [1, 3, 6, 1047, 6282]]

new a_4=6619;next state=[1047, 6619];drop ONLY oldest state entry then append new scalar;prefix now=[19, 21, 167, 1047, 6619]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 1047)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 2094

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(3, 2094)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 2097

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(6, 6619)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 39714

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(2097, 39714)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 41811

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[1047, 6619];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 1047, 2094], [1, 4, 6, 6619, 39714]]

new a_5=41811;next state=[6619, 41811];drop ONLY oldest state entry then append new scalar;prefix now=[19, 21, 167, 1047, 6619, 41811]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 6619)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 13238

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(3, 13238)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 13241

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(6, 41811)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 250866

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(13241, 250866)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 264107

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[6619, 41811];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 6619, 13238], [1, 5, 6, 41811, 250866]]

new a_6=264107;next state=[41811, 264107];drop ONLY oldest state entry then append new scalar;prefix now=[19, 21, 167, 1047, 6619, 41811, 264107]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 41811)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 83622

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(3, 83622)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 83625

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(6, 264107)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 1584642

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(83625, 1584642)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 1668267

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[41811, 264107];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 41811, 83622], [1, 6, 6, 264107, 1584642]]

new a_7=1668267;next state=[264107, 1668267];drop ONLY oldest state entry then append new scalar;prefix now=[19, 21, 167, 1047, 6619, 41811, 264107, 1668267]

recurrence RIGHT completion:complete ORIGINAL initial state=[19, 21];actual generated prefix=[19, 21, 167, 1047, 6619, 41811, 264107, 1668267]

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=19;different completions a_1=20 and21;target=7

WHOLE prefixes including initials differ=Yes;actual targets=[1607779, 1668267] 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

(19, 20, 21, [19, 20, 161, 1009, 6379, 40295, 254531, 1607779], [19, 21, 167, 1047, 6619, 41811, 264107, 1668267], 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.