第 22571 題：遞迴關係

原題與已知條件

initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={"coefficients":[3,2],"constant":0,"first_initial":20,"second_left":21,"second_right":22,"target_n":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — initial_values[j]=a_j且coefficients[j]乘a_(n-r+j)的明示order-r state update。操作=recurrence-missing-initial-boundary；資料={"coefficients":[3,2],"constant":0,"first_initial":20,"second_left":21,"second_right":22,"target_n":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-recurrence-relation

逐步推導

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=0;ALL initial a_0..a_(r-1)=[20, 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 LEFT completion n=2 ORIGINAL coefficient0 times a_0 := mul(3, 20)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 60

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(0, 60)

recurrence LEFT completion n=2 partial sum after coefficient0 = 60

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

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 42

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(60, 42)

recurrence LEFT completion n=2 partial sum after coefficient1 = 102

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[20, 21];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 20, 60], [1, 1, 2, 21, 42]]

new a_2=102;next state=[21, 102];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 102]

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

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 63

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(0, 63)

recurrence LEFT completion n=3 partial sum after coefficient0 = 63

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(2, 102)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 204

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(63, 204)

recurrence LEFT completion n=3 partial sum after coefficient1 = 267

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[21, 102];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 21, 63], [1, 2, 2, 102, 204]]

new a_3=267;next state=[102, 267];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 102, 267]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 102)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 306

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(0, 306)

recurrence LEFT completion n=4 partial sum after coefficient0 = 306

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(2, 267)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 534

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(306, 534)

recurrence LEFT completion n=4 partial sum after coefficient1 = 840

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[102, 267];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 102, 306], [1, 3, 2, 267, 534]]

new a_4=840;next state=[267, 840];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 102, 267, 840]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 267)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 801

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(0, 801)

recurrence LEFT completion n=5 partial sum after coefficient0 = 801

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(2, 840)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 1680

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(801, 1680)

recurrence LEFT completion n=5 partial sum after coefficient1 = 2481

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[267, 840];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 267, 801], [1, 4, 2, 840, 1680]]

new a_5=2481;next state=[840, 2481];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 102, 267, 840, 2481]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 840)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 2520

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(0, 2520)

recurrence LEFT completion n=6 partial sum after coefficient0 = 2520

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(2, 2481)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 4962

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(2520, 4962)

recurrence LEFT completion n=6 partial sum after coefficient1 = 7482

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[840, 2481];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 840, 2520], [1, 5, 2, 2481, 4962]]

new a_6=7482;next state=[2481, 7482];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 102, 267, 840, 2481, 7482]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 2481)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 7443

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(0, 7443)

recurrence LEFT completion n=7 partial sum after coefficient0 = 7443

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(2, 7482)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 14964

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(7443, 14964)

recurrence LEFT completion n=7 partial sum after coefficient1 = 22407

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[2481, 7482];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 2481, 7443], [1, 6, 2, 7482, 14964]]

new a_7=22407;next state=[7482, 22407];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 102, 267, 840, 2481, 7482, 22407]

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 7482)

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 = 22446

recurrence LEFT completion n=8 partial sum after coefficient0 := sum(0, 22446)

recurrence LEFT completion n=8 partial sum after coefficient0 = 22446

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 := mul(2, 22407)

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 = 44814

recurrence LEFT completion n=8 partial sum after coefficient1 := sum(22446, 44814)

recurrence LEFT completion n=8 partial sum after coefficient1 = 67260

recurrence LEFT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[7482, 22407];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 7482, 22446], [1, 7, 2, 22407, 44814]]

new a_8=67260;next state=[22407, 67260];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 102, 267, 840, 2481, 7482, 22407, 67260]

recurrence LEFT completion:complete ORIGINAL initial state=[20, 21];actual generated prefix=[20, 21, 102, 267, 840, 2481, 7482, 22407, 67260]

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=0;ALL initial a_0..a_(r-1)=[20, 22]

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, 20)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 60

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(0, 60)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 60

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(2, 22)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 44

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(60, 44)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 104

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[20, 22];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 20, 60], [1, 1, 2, 22, 44]]

new a_2=104;next state=[22, 104];drop ONLY oldest state entry then append new scalar;prefix now=[20, 22, 104]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 22)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 66

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(0, 66)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 66

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(2, 104)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 208

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(66, 208)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 274

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[22, 104];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 22, 66], [1, 2, 2, 104, 208]]

new a_3=274;next state=[104, 274];drop ONLY oldest state entry then append new scalar;prefix now=[20, 22, 104, 274]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 104)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 312

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(0, 312)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 312

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(2, 274)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 548

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(312, 548)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 860

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[104, 274];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 104, 312], [1, 3, 2, 274, 548]]

new a_4=860;next state=[274, 860];drop ONLY oldest state entry then append new scalar;prefix now=[20, 22, 104, 274, 860]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 274)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 822

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(0, 822)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 822

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(2, 860)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 1720

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(822, 1720)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 2542

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[274, 860];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 274, 822], [1, 4, 2, 860, 1720]]

new a_5=2542;next state=[860, 2542];drop ONLY oldest state entry then append new scalar;prefix now=[20, 22, 104, 274, 860, 2542]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 860)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 2580

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(0, 2580)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 2580

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(2, 2542)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 5084

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(2580, 5084)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 7664

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[860, 2542];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 860, 2580], [1, 5, 2, 2542, 5084]]

new a_6=7664;next state=[2542, 7664];drop ONLY oldest state entry then append new scalar;prefix now=[20, 22, 104, 274, 860, 2542, 7664]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 2542)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 7626

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(0, 7626)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 7626

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(2, 7664)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 15328

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(7626, 15328)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 22954

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[2542, 7664];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 2542, 7626], [1, 6, 2, 7664, 15328]]

new a_7=22954;next state=[7664, 22954];drop ONLY oldest state entry then append new scalar;prefix now=[20, 22, 104, 274, 860, 2542, 7664, 22954]

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 7664)

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 = 22992

recurrence RIGHT completion n=8 partial sum after coefficient0 := sum(0, 22992)

recurrence RIGHT completion n=8 partial sum after coefficient0 = 22992

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 := mul(2, 22954)

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 = 45908

recurrence RIGHT completion n=8 partial sum after coefficient1 := sum(22992, 45908)

recurrence RIGHT completion n=8 partial sum after coefficient1 = 68900

recurrence RIGHT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[7664, 22954];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 7664, 22992], [1, 7, 2, 22954, 45908]]

new a_8=68900;next state=[22954, 68900];drop ONLY oldest state entry then append new scalar;prefix now=[20, 22, 104, 274, 860, 2542, 7664, 22954, 68900]

recurrence RIGHT completion:complete ORIGINAL initial state=[20, 22];actual generated prefix=[20, 22, 104, 274, 860, 2542, 7664, 22954, 68900]

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=20;different completions a_1=21 and22;target=8

WHOLE prefixes including initials differ=是;actual targets=[67260, 68900] MAY coincide;one_initial_uniqueness_claim fixed No

逐步反向核對

• S076 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S075 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S074 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S073 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=8 partial sum after coefficient1，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=8 partial sum after coefficient0，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6，不以小數近似替代。
• S069 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S068 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=7 partial sum after coefficient1，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=7 partial sum after coefficient0，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5，不以小數近似替代。
• S064 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S063 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=6 partial sum after coefficient1，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=6 partial sum after coefficient0，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4，不以小數近似替代。
• S059 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S058 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=5 partial sum after coefficient1，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=5 partial sum after coefficient0，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3，不以小數近似替代。
• S054 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S053 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=4 partial sum after coefficient1，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=4 partial sum after coefficient0，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2，不以小數近似替代。
• S049 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S048 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=3 partial sum after coefficient1，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=3 partial sum after coefficient0，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1，不以小數近似替代。
• S044 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S043 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=2 partial sum after coefficient1，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=2 partial sum after coefficient0，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0，不以小數近似替代。
• S039 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S038 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S037 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S036 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=8 partial sum after coefficient1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=8 partial sum after coefficient0，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6，不以小數近似替代。
• S032 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S031 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=7 partial sum after coefficient1，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=7 partial sum after coefficient0，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5，不以小數近似替代。
• S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S026 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=6 partial sum after coefficient1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=6 partial sum after coefficient0，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4，不以小數近似替代。
• S022 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S021 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=5 partial sum after coefficient1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=5 partial sum after coefficient0，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3，不以小數近似替代。
• S017 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S016 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=4 partial sum after coefficient1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=4 partial sum after coefficient0，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2，不以小數近似替代。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=3 partial sum after coefficient1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=3 partial sum after coefficient0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1，不以小數近似替代。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=2 partial sum after coefficient1，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=2 partial sum after coefficient0，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(20, 21, 22, [20, 21, 102, 267, 840, 2481, 7482, 22407, 67260], [20, 22, 104, 274, 860, 2542, 7664, 22954, 68900], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。