第 22564 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":[2,5],"constant":1,"first_initial":13,"second_left":14,"second_right":15,"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":[2,5],"constant":1,"first_initial":13,"second_left":14,"second_right":15,"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, 5];constant=1;ALL initial a_0..a_(r-1)=[13, 14]

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, 13)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 26

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(1, 26)

recurrence LEFT completion n=2 partial sum after coefficient0 = 27

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(5, 14)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 70

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(27, 70)

recurrence LEFT completion n=2 partial sum after coefficient1 = 97

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[13, 14];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 13, 26], [1, 1, 5, 14, 70]]

new a_2=97;next state=[14, 97];drop ONLY oldest state entry then append new scalar;prefix now=[13, 14, 97]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 14)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 28

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(1, 28)

recurrence LEFT completion n=3 partial sum after coefficient0 = 29

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(5, 97)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 485

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(29, 485)

recurrence LEFT completion n=3 partial sum after coefficient1 = 514

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[14, 97];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 14, 28], [1, 2, 5, 97, 485]]

new a_3=514;next state=[97, 514];drop ONLY oldest state entry then append new scalar;prefix now=[13, 14, 97, 514]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 97)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 194

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(1, 194)

recurrence LEFT completion n=4 partial sum after coefficient0 = 195

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(5, 514)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 2570

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(195, 2570)

recurrence LEFT completion n=4 partial sum after coefficient1 = 2765

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[97, 514];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 97, 194], [1, 3, 5, 514, 2570]]

new a_4=2765;next state=[514, 2765];drop ONLY oldest state entry then append new scalar;prefix now=[13, 14, 97, 514, 2765]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 514)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 1028

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(1, 1028)

recurrence LEFT completion n=5 partial sum after coefficient0 = 1029

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(5, 2765)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 13825

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(1029, 13825)

recurrence LEFT completion n=5 partial sum after coefficient1 = 14854

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[514, 2765];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 514, 1028], [1, 4, 5, 2765, 13825]]

new a_5=14854;next state=[2765, 14854];drop ONLY oldest state entry then append new scalar;prefix now=[13, 14, 97, 514, 2765, 14854]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 2765)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 5530

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(1, 5530)

recurrence LEFT completion n=6 partial sum after coefficient0 = 5531

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(5, 14854)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 74270

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(5531, 74270)

recurrence LEFT completion n=6 partial sum after coefficient1 = 79801

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[2765, 14854];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 2765, 5530], [1, 5, 5, 14854, 74270]]

new a_6=79801;next state=[14854, 79801];drop ONLY oldest state entry then append new scalar;prefix now=[13, 14, 97, 514, 2765, 14854, 79801]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 14854)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 29708

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(1, 29708)

recurrence LEFT completion n=7 partial sum after coefficient0 = 29709

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(5, 79801)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 399005

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(29709, 399005)

recurrence LEFT completion n=7 partial sum after coefficient1 = 428714

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[14854, 79801];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 14854, 29708], [1, 6, 5, 79801, 399005]]

new a_7=428714;next state=[79801, 428714];drop ONLY oldest state entry then append new scalar;prefix now=[13, 14, 97, 514, 2765, 14854, 79801, 428714]

recurrence LEFT completion:complete ORIGINAL initial state=[13, 14];actual generated prefix=[13, 14, 97, 514, 2765, 14854, 79801, 428714]

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, 5];constant=1;ALL initial a_0..a_(r-1)=[13, 15]

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, 13)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 26

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(1, 26)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 27

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(5, 15)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 75

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(27, 75)

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

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[13, 15];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 13, 26], [1, 1, 5, 15, 75]]

new a_2=102;next state=[15, 102];drop ONLY oldest state entry then append new scalar;prefix now=[13, 15, 102]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 15)

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

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(1, 30)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 31

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(5, 102)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 510

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(31, 510)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 541

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[15, 102];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 15, 30], [1, 2, 5, 102, 510]]

new a_3=541;next state=[102, 541];drop ONLY oldest state entry then append new scalar;prefix now=[13, 15, 102, 541]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 102)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 204

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(1, 204)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 205

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(5, 541)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 2705

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(205, 2705)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 2910

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[102, 541];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 102, 204], [1, 3, 5, 541, 2705]]

new a_4=2910;next state=[541, 2910];drop ONLY oldest state entry then append new scalar;prefix now=[13, 15, 102, 541, 2910]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 541)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 1082

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(1, 1082)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 1083

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(5, 2910)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 14550

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(1083, 14550)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 15633

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[541, 2910];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 541, 1082], [1, 4, 5, 2910, 14550]]

new a_5=15633;next state=[2910, 15633];drop ONLY oldest state entry then append new scalar;prefix now=[13, 15, 102, 541, 2910, 15633]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 2910)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 5820

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(1, 5820)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 5821

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(5, 15633)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 78165

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(5821, 78165)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 83986

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[2910, 15633];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 2910, 5820], [1, 5, 5, 15633, 78165]]

new a_6=83986;next state=[15633, 83986];drop ONLY oldest state entry then append new scalar;prefix now=[13, 15, 102, 541, 2910, 15633, 83986]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 15633)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 31266

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(1, 31266)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 31267

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(5, 83986)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 419930

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(31267, 419930)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 451197

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[15633, 83986];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 15633, 31266], [1, 6, 5, 83986, 419930]]

new a_7=451197;next state=[83986, 451197];drop ONLY oldest state entry then append new scalar;prefix now=[13, 15, 102, 541, 2910, 15633, 83986, 451197]

recurrence RIGHT completion:complete ORIGINAL initial state=[13, 15];actual generated prefix=[13, 15, 102, 541, 2910, 15633, 83986, 451197]

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=13;different completions a_1=14 and15;target=7

WHOLE prefixes including initials differ=是;actual targets=[428714, 451197] 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

(13, 14, 15, [13, 14, 97, 514, 2765, 14854, 79801, 428714], [13, 15, 102, 541, 2910, 15633, 83986, 451197], 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。