第 22555 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":[2,6],"constant":0,"first_initial":4,"second_left":5,"second_right":6,"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":[2,6],"constant":0,"first_initial":4,"second_left":5,"second_right":6,"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, 6];constant=0;ALL initial a_0..a_(r-1)=[4, 5]

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, 4)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 8

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(0, 8)

recurrence LEFT completion n=2 partial sum after coefficient0 = 8

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(6, 5)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 30

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(8, 30)

recurrence LEFT completion n=2 partial sum after coefficient1 = 38

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[4, 5];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 4, 8], [1, 1, 6, 5, 30]]

new a_2=38;next state=[5, 38];drop ONLY oldest state entry then append new scalar;prefix now=[4, 5, 38]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 5)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 10

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(0, 10)

recurrence LEFT completion n=3 partial sum after coefficient0 = 10

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(6, 38)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 228

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(10, 228)

recurrence LEFT completion n=3 partial sum after coefficient1 = 238

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[5, 38];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 5, 10], [1, 2, 6, 38, 228]]

new a_3=238;next state=[38, 238];drop ONLY oldest state entry then append new scalar;prefix now=[4, 5, 38, 238]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 38)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 76

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(0, 76)

recurrence LEFT completion n=4 partial sum after coefficient0 = 76

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(6, 238)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 1428

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(76, 1428)

recurrence LEFT completion n=4 partial sum after coefficient1 = 1504

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[38, 238];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 38, 76], [1, 3, 6, 238, 1428]]

new a_4=1504;next state=[238, 1504];drop ONLY oldest state entry then append new scalar;prefix now=[4, 5, 38, 238, 1504]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 238)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 476

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(0, 476)

recurrence LEFT completion n=5 partial sum after coefficient0 = 476

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(6, 1504)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 9024

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(476, 9024)

recurrence LEFT completion n=5 partial sum after coefficient1 = 9500

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[238, 1504];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 238, 476], [1, 4, 6, 1504, 9024]]

new a_5=9500;next state=[1504, 9500];drop ONLY oldest state entry then append new scalar;prefix now=[4, 5, 38, 238, 1504, 9500]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 1504)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 3008

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(0, 3008)

recurrence LEFT completion n=6 partial sum after coefficient0 = 3008

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(6, 9500)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 57000

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(3008, 57000)

recurrence LEFT completion n=6 partial sum after coefficient1 = 60008

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1504, 9500];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 1504, 3008], [1, 5, 6, 9500, 57000]]

new a_6=60008;next state=[9500, 60008];drop ONLY oldest state entry then append new scalar;prefix now=[4, 5, 38, 238, 1504, 9500, 60008]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 9500)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 19000

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(0, 19000)

recurrence LEFT completion n=7 partial sum after coefficient0 = 19000

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(6, 60008)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 360048

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(19000, 360048)

recurrence LEFT completion n=7 partial sum after coefficient1 = 379048

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[9500, 60008];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 9500, 19000], [1, 6, 6, 60008, 360048]]

new a_7=379048;next state=[60008, 379048];drop ONLY oldest state entry then append new scalar;prefix now=[4, 5, 38, 238, 1504, 9500, 60008, 379048]

recurrence LEFT completion:complete ORIGINAL initial state=[4, 5];actual generated prefix=[4, 5, 38, 238, 1504, 9500, 60008, 379048]

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, 6];constant=0;ALL initial a_0..a_(r-1)=[4, 6]

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, 4)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 8

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(0, 8)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 8

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(6, 6)

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

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(8, 36)

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

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[4, 6];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 4, 8], [1, 1, 6, 6, 36]]

new a_2=44;next state=[6, 44];drop ONLY oldest state entry then append new scalar;prefix now=[4, 6, 44]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 6)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 12

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(0, 12)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 12

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(6, 44)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 264

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(12, 264)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 276

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[6, 44];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 6, 12], [1, 2, 6, 44, 264]]

new a_3=276;next state=[44, 276];drop ONLY oldest state entry then append new scalar;prefix now=[4, 6, 44, 276]

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

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

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(0, 88)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 88

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(6, 276)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 1656

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(88, 1656)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 1744

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[44, 276];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 44, 88], [1, 3, 6, 276, 1656]]

new a_4=1744;next state=[276, 1744];drop ONLY oldest state entry then append new scalar;prefix now=[4, 6, 44, 276, 1744]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 276)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 552

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(0, 552)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 552

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(6, 1744)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 10464

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(552, 10464)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 11016

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[276, 1744];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 276, 552], [1, 4, 6, 1744, 10464]]

new a_5=11016;next state=[1744, 11016];drop ONLY oldest state entry then append new scalar;prefix now=[4, 6, 44, 276, 1744, 11016]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 1744)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 3488

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(0, 3488)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 3488

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(6, 11016)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 66096

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(3488, 66096)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 69584

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1744, 11016];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 1744, 3488], [1, 5, 6, 11016, 66096]]

new a_6=69584;next state=[11016, 69584];drop ONLY oldest state entry then append new scalar;prefix now=[4, 6, 44, 276, 1744, 11016, 69584]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 11016)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 22032

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(0, 22032)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 22032

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(6, 69584)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 417504

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(22032, 417504)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 439536

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[11016, 69584];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 11016, 22032], [1, 6, 6, 69584, 417504]]

new a_7=439536;next state=[69584, 439536];drop ONLY oldest state entry then append new scalar;prefix now=[4, 6, 44, 276, 1744, 11016, 69584, 439536]

recurrence RIGHT completion:complete ORIGINAL initial state=[4, 6];actual generated prefix=[4, 6, 44, 276, 1744, 11016, 69584, 439536]

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=4;different completions a_1=5 and6;target=7

WHOLE prefixes including initials differ=是;actual targets=[379048, 439536] 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

(4, 5, 6, [4, 5, 38, 238, 1504, 9500, 60008, 379048], [4, 6, 44, 276, 1744, 11016, 69584, 439536], 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。