第 22559 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":[3,5],"constant":0,"first_initial":8,"second_left":9,"second_right":10,"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":[3,5],"constant":0,"first_initial":8,"second_left":9,"second_right":10,"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, 5];constant=0;ALL initial a_0..a_(r-1)=[8, 9]

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, 8)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 24

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(0, 24)

recurrence LEFT completion n=2 partial sum after coefficient0 = 24

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(5, 9)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 45

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(24, 45)

recurrence LEFT completion n=2 partial sum after coefficient1 = 69

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[8, 9];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 8, 24], [1, 1, 5, 9, 45]]

new a_2=69;next state=[9, 69];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 69]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 9)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 27

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(0, 27)

recurrence LEFT completion n=3 partial sum after coefficient0 = 27

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(5, 69)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 345

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(27, 345)

recurrence LEFT completion n=3 partial sum after coefficient1 = 372

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[9, 69];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 9, 27], [1, 2, 5, 69, 345]]

new a_3=372;next state=[69, 372];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 69, 372]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 69)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 207

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(0, 207)

recurrence LEFT completion n=4 partial sum after coefficient0 = 207

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(5, 372)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 1860

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(207, 1860)

recurrence LEFT completion n=4 partial sum after coefficient1 = 2067

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[69, 372];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 69, 207], [1, 3, 5, 372, 1860]]

new a_4=2067;next state=[372, 2067];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 69, 372, 2067]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 372)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 1116

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(0, 1116)

recurrence LEFT completion n=5 partial sum after coefficient0 = 1116

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(5, 2067)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 10335

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(1116, 10335)

recurrence LEFT completion n=5 partial sum after coefficient1 = 11451

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[372, 2067];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 372, 1116], [1, 4, 5, 2067, 10335]]

new a_5=11451;next state=[2067, 11451];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 69, 372, 2067, 11451]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 2067)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 6201

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(0, 6201)

recurrence LEFT completion n=6 partial sum after coefficient0 = 6201

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(5, 11451)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 57255

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(6201, 57255)

recurrence LEFT completion n=6 partial sum after coefficient1 = 63456

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[2067, 11451];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 2067, 6201], [1, 5, 5, 11451, 57255]]

new a_6=63456;next state=[11451, 63456];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 69, 372, 2067, 11451, 63456]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 11451)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 34353

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(0, 34353)

recurrence LEFT completion n=7 partial sum after coefficient0 = 34353

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(5, 63456)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 317280

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(34353, 317280)

recurrence LEFT completion n=7 partial sum after coefficient1 = 351633

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[11451, 63456];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 11451, 34353], [1, 6, 5, 63456, 317280]]

new a_7=351633;next state=[63456, 351633];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 69, 372, 2067, 11451, 63456, 351633]

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 63456)

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 = 190368

recurrence LEFT completion n=8 partial sum after coefficient0 := sum(0, 190368)

recurrence LEFT completion n=8 partial sum after coefficient0 = 190368

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 := mul(5, 351633)

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 = 1758165

recurrence LEFT completion n=8 partial sum after coefficient1 := sum(190368, 1758165)

recurrence LEFT completion n=8 partial sum after coefficient1 = 1948533

recurrence LEFT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[63456, 351633];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 63456, 190368], [1, 7, 5, 351633, 1758165]]

new a_8=1948533;next state=[351633, 1948533];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 69, 372, 2067, 11451, 63456, 351633, 1948533]

recurrence LEFT completion:complete ORIGINAL initial state=[8, 9];actual generated prefix=[8, 9, 69, 372, 2067, 11451, 63456, 351633, 1948533]

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, 5];constant=0;ALL initial a_0..a_(r-1)=[8, 10]

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, 8)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 24

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(0, 24)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 24

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(5, 10)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 50

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(24, 50)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 74

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[8, 10];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 8, 24], [1, 1, 5, 10, 50]]

new a_2=74;next state=[10, 74];drop ONLY oldest state entry then append new scalar;prefix now=[8, 10, 74]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 10)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 30

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(0, 30)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 30

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(5, 74)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 370

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(30, 370)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 400

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[10, 74];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 10, 30], [1, 2, 5, 74, 370]]

new a_3=400;next state=[74, 400];drop ONLY oldest state entry then append new scalar;prefix now=[8, 10, 74, 400]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 74)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 222

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(0, 222)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 222

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(5, 400)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 2000

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(222, 2000)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 2222

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[74, 400];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 74, 222], [1, 3, 5, 400, 2000]]

new a_4=2222;next state=[400, 2222];drop ONLY oldest state entry then append new scalar;prefix now=[8, 10, 74, 400, 2222]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 400)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 1200

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(0, 1200)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 1200

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(5, 2222)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 11110

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(1200, 11110)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 12310

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[400, 2222];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 400, 1200], [1, 4, 5, 2222, 11110]]

new a_5=12310;next state=[2222, 12310];drop ONLY oldest state entry then append new scalar;prefix now=[8, 10, 74, 400, 2222, 12310]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 2222)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 6666

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(0, 6666)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 6666

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(5, 12310)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 61550

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(6666, 61550)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 68216

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[2222, 12310];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 2222, 6666], [1, 5, 5, 12310, 61550]]

new a_6=68216;next state=[12310, 68216];drop ONLY oldest state entry then append new scalar;prefix now=[8, 10, 74, 400, 2222, 12310, 68216]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 12310)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 36930

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(0, 36930)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 36930

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(5, 68216)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 341080

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(36930, 341080)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 378010

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[12310, 68216];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 12310, 36930], [1, 6, 5, 68216, 341080]]

new a_7=378010;next state=[68216, 378010];drop ONLY oldest state entry then append new scalar;prefix now=[8, 10, 74, 400, 2222, 12310, 68216, 378010]

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 68216)

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 = 204648

recurrence RIGHT completion n=8 partial sum after coefficient0 := sum(0, 204648)

recurrence RIGHT completion n=8 partial sum after coefficient0 = 204648

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 := mul(5, 378010)

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 = 1890050

recurrence RIGHT completion n=8 partial sum after coefficient1 := sum(204648, 1890050)

recurrence RIGHT completion n=8 partial sum after coefficient1 = 2094698

recurrence RIGHT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[68216, 378010];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 68216, 204648], [1, 7, 5, 378010, 1890050]]

new a_8=2094698;next state=[378010, 2094698];drop ONLY oldest state entry then append new scalar;prefix now=[8, 10, 74, 400, 2222, 12310, 68216, 378010, 2094698]

recurrence RIGHT completion:complete ORIGINAL initial state=[8, 10];actual generated prefix=[8, 10, 74, 400, 2222, 12310, 68216, 378010, 2094698]

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=8;different completions a_1=9 and10;target=8

WHOLE prefixes including initials differ=是;actual targets=[1948533, 2094698] 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

(8, 9, 10, [8, 9, 69, 372, 2067, 11451, 63456, 351633, 1948533], [8, 10, 74, 400, 2222, 12310, 68216, 378010, 2094698], 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。