第 22562 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":[3,3],"constant":3,"first_initial":11,"second_left":12,"second_right":13,"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":[3,3],"constant":3,"first_initial":11,"second_left":12,"second_right":13,"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, 3];constant=3;ALL initial a_0..a_(r-1)=[11, 12]

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, 11)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 33

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(3, 33)

recurrence LEFT completion n=2 partial sum after coefficient0 = 36

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 12)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 36

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(36, 36)

recurrence LEFT completion n=2 partial sum after coefficient1 = 72

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[11, 12];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 11, 33], [1, 1, 3, 12, 36]]

new a_2=72;next state=[12, 72];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 12)

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

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

recurrence LEFT completion n=3 partial sum after coefficient0 = 39

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 72)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 216

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(39, 216)

recurrence LEFT completion n=3 partial sum after coefficient1 = 255

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[12, 72];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 12, 36], [1, 2, 3, 72, 216]]

new a_3=255;next state=[72, 255];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 72)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 216

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(3, 216)

recurrence LEFT completion n=4 partial sum after coefficient0 = 219

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 255)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 765

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(219, 765)

recurrence LEFT completion n=4 partial sum after coefficient1 = 984

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[72, 255];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 72, 216], [1, 3, 3, 255, 765]]

new a_4=984;next state=[255, 984];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255, 984]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 255)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 765

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(3, 765)

recurrence LEFT completion n=5 partial sum after coefficient0 = 768

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 984)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 2952

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(768, 2952)

recurrence LEFT completion n=5 partial sum after coefficient1 = 3720

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[255, 984];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 255, 765], [1, 4, 3, 984, 2952]]

new a_5=3720;next state=[984, 3720];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255, 984, 3720]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 984)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 2952

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(3, 2952)

recurrence LEFT completion n=6 partial sum after coefficient0 = 2955

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 3720)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 11160

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(2955, 11160)

recurrence LEFT completion n=6 partial sum after coefficient1 = 14115

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[984, 3720];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 984, 2952], [1, 5, 3, 3720, 11160]]

new a_6=14115;next state=[3720, 14115];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255, 984, 3720, 14115]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 3720)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 11160

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(3, 11160)

recurrence LEFT completion n=7 partial sum after coefficient0 = 11163

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(3, 14115)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 42345

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(11163, 42345)

recurrence LEFT completion n=7 partial sum after coefficient1 = 53508

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[3720, 14115];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 3720, 11160], [1, 6, 3, 14115, 42345]]

new a_7=53508;next state=[14115, 53508];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255, 984, 3720, 14115, 53508]

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 14115)

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 = 42345

recurrence LEFT completion n=8 partial sum after coefficient0 := sum(3, 42345)

recurrence LEFT completion n=8 partial sum after coefficient0 = 42348

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 := mul(3, 53508)

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 = 160524

recurrence LEFT completion n=8 partial sum after coefficient1 := sum(42348, 160524)

recurrence LEFT completion n=8 partial sum after coefficient1 = 202872

recurrence LEFT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[14115, 53508];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 14115, 42345], [1, 7, 3, 53508, 160524]]

new a_8=202872;next state=[53508, 202872];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255, 984, 3720, 14115, 53508, 202872]

recurrence LEFT completion:complete ORIGINAL initial state=[11, 12];actual generated prefix=[11, 12, 72, 255, 984, 3720, 14115, 53508, 202872]

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, 3];constant=3;ALL initial a_0..a_(r-1)=[11, 13]

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, 11)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 33

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(3, 33)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 36

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 13)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 39

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(36, 39)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 75

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[11, 13];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 11, 33], [1, 1, 3, 13, 39]]

new a_2=75;next state=[13, 75];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 13)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 39

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(3, 39)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 42

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 75)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 225

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(42, 225)

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

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[13, 75];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 13, 39], [1, 2, 3, 75, 225]]

new a_3=267;next state=[75, 267];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 75)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 225

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(3, 225)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 228

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 267)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 801

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(228, 801)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 1029

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[75, 267];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 75, 225], [1, 3, 3, 267, 801]]

new a_4=1029;next state=[267, 1029];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267, 1029]

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

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

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(3, 801)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 804

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 1029)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 3087

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(804, 3087)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 3891

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[267, 1029];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 267, 801], [1, 4, 3, 1029, 3087]]

new a_5=3891;next state=[1029, 3891];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267, 1029, 3891]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 1029)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 3087

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(3, 3087)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 3090

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 3891)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 11673

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(3090, 11673)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 14763

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1029, 3891];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 1029, 3087], [1, 5, 3, 3891, 11673]]

new a_6=14763;next state=[3891, 14763];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267, 1029, 3891, 14763]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 3891)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 11673

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(3, 11673)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 11676

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(3, 14763)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 44289

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(11676, 44289)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 55965

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[3891, 14763];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 3891, 11673], [1, 6, 3, 14763, 44289]]

new a_7=55965;next state=[14763, 55965];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267, 1029, 3891, 14763, 55965]

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 14763)

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 = 44289

recurrence RIGHT completion n=8 partial sum after coefficient0 := sum(3, 44289)

recurrence RIGHT completion n=8 partial sum after coefficient0 = 44292

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 := mul(3, 55965)

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 = 167895

recurrence RIGHT completion n=8 partial sum after coefficient1 := sum(44292, 167895)

recurrence RIGHT completion n=8 partial sum after coefficient1 = 212187

recurrence RIGHT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[14763, 55965];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 14763, 44289], [1, 7, 3, 55965, 167895]]

new a_8=212187;next state=[55965, 212187];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187]

recurrence RIGHT completion:complete ORIGINAL initial state=[11, 13];actual generated prefix=[11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187]

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=11;different completions a_1=12 and13;target=8

WHOLE prefixes including initials differ=是;actual targets=[202872, 212187] 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

(11, 12, 13, [11, 12, 72, 255, 984, 3720, 14115, 53508, 202872], [11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187], 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。