第 22561 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":[2,2],"constant":2,"first_initial":10,"second_left":11,"second_right":12,"target_n":7}。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":[2,2],"constant":2,"first_initial":10,"second_left":11,"second_right":12,"target_n":7}。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=[2, 2];constant=2;ALL initial a_0..a_(r-1)=[10, 11]

for EVERY n>=2,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=2;forward rule is total and single-valued

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 := mul(2, 10)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 20

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(2, 20)

recurrence LEFT completion n=2 partial sum after coefficient0 = 22

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(2, 11)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 22

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(22, 22)

recurrence LEFT completion n=2 partial sum after coefficient1 = 44

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[10, 11];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 10, 20], [1, 1, 2, 11, 22]]

new a_2=44;next state=[11, 44];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 44]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 11)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 22

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(2, 22)

recurrence LEFT completion n=3 partial sum after coefficient0 = 24

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(2, 44)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 88

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(24, 88)

recurrence LEFT completion n=3 partial sum after coefficient1 = 112

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[11, 44];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 11, 22], [1, 2, 2, 44, 88]]

new a_3=112;next state=[44, 112];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 44, 112]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 44)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 88

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(2, 88)

recurrence LEFT completion n=4 partial sum after coefficient0 = 90

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(2, 112)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 224

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(90, 224)

recurrence LEFT completion n=4 partial sum after coefficient1 = 314

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[44, 112];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 44, 88], [1, 3, 2, 112, 224]]

new a_4=314;next state=[112, 314];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 44, 112, 314]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 112)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 224

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(2, 224)

recurrence LEFT completion n=5 partial sum after coefficient0 = 226

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(2, 314)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 628

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(226, 628)

recurrence LEFT completion n=5 partial sum after coefficient1 = 854

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[112, 314];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 112, 224], [1, 4, 2, 314, 628]]

new a_5=854;next state=[314, 854];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 44, 112, 314, 854]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 314)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 628

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(2, 628)

recurrence LEFT completion n=6 partial sum after coefficient0 = 630

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(2, 854)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 1708

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(630, 1708)

recurrence LEFT completion n=6 partial sum after coefficient1 = 2338

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[314, 854];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 314, 628], [1, 5, 2, 854, 1708]]

new a_6=2338;next state=[854, 2338];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 44, 112, 314, 854, 2338]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 854)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 1708

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(2, 1708)

recurrence LEFT completion n=7 partial sum after coefficient0 = 1710

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(2, 2338)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 4676

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(1710, 4676)

recurrence LEFT completion n=7 partial sum after coefficient1 = 6386

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[854, 2338];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 854, 1708], [1, 6, 2, 2338, 4676]]

new a_7=6386;next state=[2338, 6386];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 44, 112, 314, 854, 2338, 6386]

recurrence LEFT completion:complete ORIGINAL initial state=[10, 11];actual generated prefix=[10, 11, 44, 112, 314, 854, 2338, 6386]

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

recurrence RIGHT completion:ORIGINAL order=2;oldest-to-newest coefficients=[2, 2];constant=2;ALL initial a_0..a_(r-1)=[10, 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 RIGHT completion n=2 ORIGINAL coefficient0 times a_0 := mul(2, 10)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 20

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(2, 20)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 22

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

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 24

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(22, 24)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 46

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[10, 12];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 10, 20], [1, 1, 2, 12, 24]]

new a_2=46;next state=[12, 46];drop ONLY oldest state entry then append new scalar;prefix now=[10, 12, 46]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 12)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 24

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(2, 24)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 26

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(2, 46)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 92

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(26, 92)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 118

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[12, 46];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 12, 24], [1, 2, 2, 46, 92]]

new a_3=118;next state=[46, 118];drop ONLY oldest state entry then append new scalar;prefix now=[10, 12, 46, 118]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 46)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 92

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(2, 92)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 94

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(2, 118)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 236

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(94, 236)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 330

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[46, 118];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 46, 92], [1, 3, 2, 118, 236]]

new a_4=330;next state=[118, 330];drop ONLY oldest state entry then append new scalar;prefix now=[10, 12, 46, 118, 330]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 118)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 236

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(2, 236)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 238

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(2, 330)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 660

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(238, 660)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 898

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[118, 330];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 118, 236], [1, 4, 2, 330, 660]]

new a_5=898;next state=[330, 898];drop ONLY oldest state entry then append new scalar;prefix now=[10, 12, 46, 118, 330, 898]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 330)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 660

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(2, 660)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 662

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(2, 898)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 1796

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(662, 1796)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 2458

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[330, 898];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 330, 660], [1, 5, 2, 898, 1796]]

new a_6=2458;next state=[898, 2458];drop ONLY oldest state entry then append new scalar;prefix now=[10, 12, 46, 118, 330, 898, 2458]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 898)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 1796

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(2, 1796)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 1798

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(2, 2458)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 4916

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(1798, 4916)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 6714

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[898, 2458];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 898, 1796], [1, 6, 2, 2458, 4916]]

new a_7=6714;next state=[2458, 6714];drop ONLY oldest state entry then append new scalar;prefix now=[10, 12, 46, 118, 330, 898, 2458, 6714]

recurrence RIGHT completion:complete ORIGINAL initial state=[10, 12];actual generated prefix=[10, 12, 46, 118, 330, 898, 2458, 6714]

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=10;different completions a_1=11 and12;target=7

WHOLE prefixes including initials differ=是;actual targets=[6386, 6714] MAY coincide;one_initial_uniqueness_claim fixed No

逐步反向核對

• S066 · 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。
• S065 · 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。
• 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=7 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=7 ORIGINAL coefficient1 times a_6，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=7 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=7 ORIGINAL coefficient0 times a_5，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=6 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=6 ORIGINAL coefficient1 times a_5，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=6 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=6 ORIGINAL coefficient0 times a_4，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=5 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=5 ORIGINAL coefficient1 times a_4，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=5 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=5 ORIGINAL coefficient0 times a_3，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=4 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=4 ORIGINAL coefficient1 times a_3，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=4 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=4 ORIGINAL coefficient0 times a_2，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=3 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=3 ORIGINAL coefficient1 times a_2，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=3 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=3 ORIGINAL coefficient0 times a_1，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 — 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。
• S037 · 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。
• S036 · 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。
• S035 · 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。
• S034 · 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。
• S033 · 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。
• 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

(10, 11, 12, [10, 11, 44, 112, 314, 854, 2338, 6386], [10, 12, 46, 118, 330, 898, 2458, 6714], 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。