Question 22567: Different Missing-Initial Completions Give Different Whole Prefixes

Original question and givens

Use order2 coefficients=[2, 3] oldest-to-newest, constant=0, known a_0=16, alternative a_1=17 or 18, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

• G001 — Use order2 coefficients=[2, 3] oldest-to-newest, constant=0, known a_0=16, alternative a_1=17 or 18, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-recurrence-relation

Step-by-step derivation

source operation=recurrence-missing-initial-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary]

recurrence LEFT completion:ORIGINAL order=2;oldest-to-newest coefficients=[2, 3];constant=0;ALL initial a_0..a_(r-1)=[16, 17]

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, 16)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 32

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(0, 32)

recurrence LEFT completion n=2 partial sum after coefficient0 = 32

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 17)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 51

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(32, 51)

recurrence LEFT completion n=2 partial sum after coefficient1 = 83

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[16, 17];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 16, 32], [1, 1, 3, 17, 51]]

new a_2=83;next state=[17, 83];drop ONLY oldest state entry then append new scalar;prefix now=[16, 17, 83]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 17)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 34

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(0, 34)

recurrence LEFT completion n=3 partial sum after coefficient0 = 34

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 83)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 249

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(34, 249)

recurrence LEFT completion n=3 partial sum after coefficient1 = 283

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[17, 83];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 17, 34], [1, 2, 3, 83, 249]]

new a_3=283;next state=[83, 283];drop ONLY oldest state entry then append new scalar;prefix now=[16, 17, 83, 283]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 83)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 166

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(0, 166)

recurrence LEFT completion n=4 partial sum after coefficient0 = 166

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 283)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 849

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(166, 849)

recurrence LEFT completion n=4 partial sum after coefficient1 = 1015

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[83, 283];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 83, 166], [1, 3, 3, 283, 849]]

new a_4=1015;next state=[283, 1015];drop ONLY oldest state entry then append new scalar;prefix now=[16, 17, 83, 283, 1015]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 283)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 566

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(0, 566)

recurrence LEFT completion n=5 partial sum after coefficient0 = 566

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 1015)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 3045

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(566, 3045)

recurrence LEFT completion n=5 partial sum after coefficient1 = 3611

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[283, 1015];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 283, 566], [1, 4, 3, 1015, 3045]]

new a_5=3611;next state=[1015, 3611];drop ONLY oldest state entry then append new scalar;prefix now=[16, 17, 83, 283, 1015, 3611]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 1015)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 2030

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(0, 2030)

recurrence LEFT completion n=6 partial sum after coefficient0 = 2030

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 3611)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 10833

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(2030, 10833)

recurrence LEFT completion n=6 partial sum after coefficient1 = 12863

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1015, 3611];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 1015, 2030], [1, 5, 3, 3611, 10833]]

new a_6=12863;next state=[3611, 12863];drop ONLY oldest state entry then append new scalar;prefix now=[16, 17, 83, 283, 1015, 3611, 12863]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 3611)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 7222

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(0, 7222)

recurrence LEFT completion n=7 partial sum after coefficient0 = 7222

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(3, 12863)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 38589

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(7222, 38589)

recurrence LEFT completion n=7 partial sum after coefficient1 = 45811

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[3611, 12863];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 3611, 7222], [1, 6, 3, 12863, 38589]]

new a_7=45811;next state=[12863, 45811];drop ONLY oldest state entry then append new scalar;prefix now=[16, 17, 83, 283, 1015, 3611, 12863, 45811]

recurrence LEFT completion:complete ORIGINAL initial state=[16, 17];actual generated prefix=[16, 17, 83, 283, 1015, 3611, 12863, 45811]

ANY two sequences with SAME r initials and SAME total rule agree at n=2 by substitution,then at every later n by induction;this determines the requested prefix only

recurrence RIGHT completion:ORIGINAL order=2;oldest-to-newest coefficients=[2, 3];constant=0;ALL initial a_0..a_(r-1)=[16, 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 RIGHT completion n=2 ORIGINAL coefficient0 times a_0 := mul(2, 16)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 32

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(0, 32)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 32

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 18)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 54

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(32, 54)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 86

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[16, 18];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 16, 32], [1, 1, 3, 18, 54]]

new a_2=86;next state=[18, 86];drop ONLY oldest state entry then append new scalar;prefix now=[16, 18, 86]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 18)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 36

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(0, 36)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 36

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 86)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 258

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(36, 258)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 294

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[18, 86];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 18, 36], [1, 2, 3, 86, 258]]

new a_3=294;next state=[86, 294];drop ONLY oldest state entry then append new scalar;prefix now=[16, 18, 86, 294]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 86)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 172

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(0, 172)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 172

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 294)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 882

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(172, 882)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 1054

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[86, 294];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 86, 172], [1, 3, 3, 294, 882]]

new a_4=1054;next state=[294, 1054];drop ONLY oldest state entry then append new scalar;prefix now=[16, 18, 86, 294, 1054]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 294)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 588

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(0, 588)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 588

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 1054)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 3162

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(588, 3162)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 3750

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[294, 1054];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 294, 588], [1, 4, 3, 1054, 3162]]

new a_5=3750;next state=[1054, 3750];drop ONLY oldest state entry then append new scalar;prefix now=[16, 18, 86, 294, 1054, 3750]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 1054)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 2108

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(0, 2108)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 2108

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 3750)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 11250

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(2108, 11250)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 13358

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1054, 3750];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 1054, 2108], [1, 5, 3, 3750, 11250]]

new a_6=13358;next state=[3750, 13358];drop ONLY oldest state entry then append new scalar;prefix now=[16, 18, 86, 294, 1054, 3750, 13358]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 3750)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 7500

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(0, 7500)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 7500

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(3, 13358)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 40074

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(7500, 40074)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 47574

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[3750, 13358];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 3750, 7500], [1, 6, 3, 13358, 40074]]

new a_7=47574;next state=[13358, 47574];drop ONLY oldest state entry then append new scalar;prefix now=[16, 18, 86, 294, 1054, 3750, 13358, 47574]

recurrence RIGHT completion:complete ORIGINAL initial state=[16, 18];actual generated prefix=[16, 18, 86, 294, 1054, 3750, 13358, 47574]

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=16;different completions a_1=17 and18;target=7

WHOLE prefixes including initials differ=Yes;actual targets=[45811, 47574] 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

(16, 17, 18, [16, 17, 83, 283, 1015, 3611, 12863, 45811], [16, 18, 86, 294, 1054, 3750, 13358, 47574], 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.