第 22574 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":3,"first_initial":23,"second_left":24,"second_right":25,"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":3,"first_initial":23,"second_left":24,"second_right":25,"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=3;ALL initial a_0..a_(r-1)=[23, 24]

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, 23)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 69

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(3, 69)

recurrence LEFT completion n=2 partial sum after coefficient0 = 72

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(5, 24)

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

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(72, 120)

recurrence LEFT completion n=2 partial sum after coefficient1 = 192

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[23, 24];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 23, 69], [1, 1, 5, 24, 120]]

new a_2=192;next state=[24, 192];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 192]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 24)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 72

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(3, 72)

recurrence LEFT completion n=3 partial sum after coefficient0 = 75

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(5, 192)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 960

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(75, 960)

recurrence LEFT completion n=3 partial sum after coefficient1 = 1035

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[24, 192];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 24, 72], [1, 2, 5, 192, 960]]

new a_3=1035;next state=[192, 1035];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 192, 1035]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 192)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 576

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(3, 576)

recurrence LEFT completion n=4 partial sum after coefficient0 = 579

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(5, 1035)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 5175

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(579, 5175)

recurrence LEFT completion n=4 partial sum after coefficient1 = 5754

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[192, 1035];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 192, 576], [1, 3, 5, 1035, 5175]]

new a_4=5754;next state=[1035, 5754];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 192, 1035, 5754]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 1035)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 3105

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(3, 3105)

recurrence LEFT completion n=5 partial sum after coefficient0 = 3108

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(5, 5754)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 28770

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(3108, 28770)

recurrence LEFT completion n=5 partial sum after coefficient1 = 31878

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[1035, 5754];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 1035, 3105], [1, 4, 5, 5754, 28770]]

new a_5=31878;next state=[5754, 31878];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 192, 1035, 5754, 31878]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 5754)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 17262

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(3, 17262)

recurrence LEFT completion n=6 partial sum after coefficient0 = 17265

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(5, 31878)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 159390

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(17265, 159390)

recurrence LEFT completion n=6 partial sum after coefficient1 = 176655

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[5754, 31878];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 5754, 17262], [1, 5, 5, 31878, 159390]]

new a_6=176655;next state=[31878, 176655];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 192, 1035, 5754, 31878, 176655]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 31878)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 95634

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(3, 95634)

recurrence LEFT completion n=7 partial sum after coefficient0 = 95637

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(5, 176655)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 883275

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(95637, 883275)

recurrence LEFT completion n=7 partial sum after coefficient1 = 978912

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[31878, 176655];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 31878, 95634], [1, 6, 5, 176655, 883275]]

new a_7=978912;next state=[176655, 978912];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 192, 1035, 5754, 31878, 176655, 978912]

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 176655)

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 = 529965

recurrence LEFT completion n=8 partial sum after coefficient0 := sum(3, 529965)

recurrence LEFT completion n=8 partial sum after coefficient0 = 529968

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 := mul(5, 978912)

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 = 4894560

recurrence LEFT completion n=8 partial sum after coefficient1 := sum(529968, 4894560)

recurrence LEFT completion n=8 partial sum after coefficient1 = 5424528

recurrence LEFT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[176655, 978912];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 176655, 529965], [1, 7, 5, 978912, 4894560]]

new a_8=5424528;next state=[978912, 5424528];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 192, 1035, 5754, 31878, 176655, 978912, 5424528]

recurrence LEFT completion:complete ORIGINAL initial state=[23, 24];actual generated prefix=[23, 24, 192, 1035, 5754, 31878, 176655, 978912, 5424528]

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=3;ALL initial a_0..a_(r-1)=[23, 25]

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, 23)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 69

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(3, 69)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 72

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(5, 25)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 125

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(72, 125)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 197

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[23, 25];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 23, 69], [1, 1, 5, 25, 125]]

new a_2=197;next state=[25, 197];drop ONLY oldest state entry then append new scalar;prefix now=[23, 25, 197]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 25)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 75

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(3, 75)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 78

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(5, 197)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 985

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(78, 985)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 1063

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[25, 197];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 25, 75], [1, 2, 5, 197, 985]]

new a_3=1063;next state=[197, 1063];drop ONLY oldest state entry then append new scalar;prefix now=[23, 25, 197, 1063]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 197)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 591

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(3, 591)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 594

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(5, 1063)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 5315

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(594, 5315)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 5909

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[197, 1063];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 197, 591], [1, 3, 5, 1063, 5315]]

new a_4=5909;next state=[1063, 5909];drop ONLY oldest state entry then append new scalar;prefix now=[23, 25, 197, 1063, 5909]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 1063)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 3189

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(3, 3189)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 3192

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(5, 5909)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 29545

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(3192, 29545)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 32737

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[1063, 5909];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 1063, 3189], [1, 4, 5, 5909, 29545]]

new a_5=32737;next state=[5909, 32737];drop ONLY oldest state entry then append new scalar;prefix now=[23, 25, 197, 1063, 5909, 32737]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 5909)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 17727

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(3, 17727)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 17730

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(5, 32737)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 163685

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(17730, 163685)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 181415

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[5909, 32737];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 5909, 17727], [1, 5, 5, 32737, 163685]]

new a_6=181415;next state=[32737, 181415];drop ONLY oldest state entry then append new scalar;prefix now=[23, 25, 197, 1063, 5909, 32737, 181415]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 32737)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 98211

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(3, 98211)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 98214

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(5, 181415)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 907075

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(98214, 907075)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 1005289

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[32737, 181415];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 32737, 98211], [1, 6, 5, 181415, 907075]]

new a_7=1005289;next state=[181415, 1005289];drop ONLY oldest state entry then append new scalar;prefix now=[23, 25, 197, 1063, 5909, 32737, 181415, 1005289]

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 181415)

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 = 544245

recurrence RIGHT completion n=8 partial sum after coefficient0 := sum(3, 544245)

recurrence RIGHT completion n=8 partial sum after coefficient0 = 544248

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 := mul(5, 1005289)

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 = 5026445

recurrence RIGHT completion n=8 partial sum after coefficient1 := sum(544248, 5026445)

recurrence RIGHT completion n=8 partial sum after coefficient1 = 5570693

recurrence RIGHT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[181415, 1005289];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 181415, 544245], [1, 7, 5, 1005289, 5026445]]

new a_8=5570693;next state=[1005289, 5570693];drop ONLY oldest state entry then append new scalar;prefix now=[23, 25, 197, 1063, 5909, 32737, 181415, 1005289, 5570693]

recurrence RIGHT completion:complete ORIGINAL initial state=[23, 25];actual generated prefix=[23, 25, 197, 1063, 5909, 32737, 181415, 1005289, 5570693]

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=23;different completions a_1=24 and25;target=8

WHOLE prefixes including initials differ=是;actual targets=[5424528, 5570693] 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

(23, 24, 25, [23, 24, 192, 1035, 5754, 31878, 176655, 978912, 5424528], [23, 25, 197, 1063, 5909, 32737, 181415, 1005289, 5570693], 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。