第 22573 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":[2,4],"constant":2,"first_initial":22,"second_left":23,"second_right":24,"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":[2,4],"constant":2,"first_initial":22,"second_left":23,"second_right":24,"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, 4];constant=2;ALL initial a_0..a_(r-1)=[22, 23]

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, 22)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 44

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(2, 44)

recurrence LEFT completion n=2 partial sum after coefficient0 = 46

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(4, 23)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 92

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(46, 92)

recurrence LEFT completion n=2 partial sum after coefficient1 = 138

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[22, 23];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 22, 44], [1, 1, 4, 23, 92]]

new a_2=138;next state=[23, 138];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 138]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 23)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 46

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(2, 46)

recurrence LEFT completion n=3 partial sum after coefficient0 = 48

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 138)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 552

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(48, 552)

recurrence LEFT completion n=3 partial sum after coefficient1 = 600

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[23, 138];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 23, 46], [1, 2, 4, 138, 552]]

new a_3=600;next state=[138, 600];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 138, 600]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 138)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 276

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(2, 276)

recurrence LEFT completion n=4 partial sum after coefficient0 = 278

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 600)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 2400

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(278, 2400)

recurrence LEFT completion n=4 partial sum after coefficient1 = 2678

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[138, 600];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 138, 276], [1, 3, 4, 600, 2400]]

new a_4=2678;next state=[600, 2678];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 138, 600, 2678]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 600)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 1200

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(2, 1200)

recurrence LEFT completion n=5 partial sum after coefficient0 = 1202

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 2678)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 10712

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(1202, 10712)

recurrence LEFT completion n=5 partial sum after coefficient1 = 11914

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[600, 2678];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 600, 1200], [1, 4, 4, 2678, 10712]]

new a_5=11914;next state=[2678, 11914];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 138, 600, 2678, 11914]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 2678)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 5356

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(2, 5356)

recurrence LEFT completion n=6 partial sum after coefficient0 = 5358

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 11914)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 47656

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(5358, 47656)

recurrence LEFT completion n=6 partial sum after coefficient1 = 53014

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[2678, 11914];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 2678, 5356], [1, 5, 4, 11914, 47656]]

new a_6=53014;next state=[11914, 53014];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 138, 600, 2678, 11914, 53014]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 11914)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 23828

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(2, 23828)

recurrence LEFT completion n=7 partial sum after coefficient0 = 23830

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(4, 53014)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 212056

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(23830, 212056)

recurrence LEFT completion n=7 partial sum after coefficient1 = 235886

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[11914, 53014];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 11914, 23828], [1, 6, 4, 53014, 212056]]

new a_7=235886;next state=[53014, 235886];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 138, 600, 2678, 11914, 53014, 235886]

recurrence LEFT completion:complete ORIGINAL initial state=[22, 23];actual generated prefix=[22, 23, 138, 600, 2678, 11914, 53014, 235886]

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, 4];constant=2;ALL initial a_0..a_(r-1)=[22, 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 RIGHT completion n=2 ORIGINAL coefficient0 times a_0 := mul(2, 22)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 44

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(2, 44)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 46

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(4, 24)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 96

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(46, 96)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 142

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[22, 24];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 22, 44], [1, 1, 4, 24, 96]]

new a_2=142;next state=[24, 142];drop ONLY oldest state entry then append new scalar;prefix now=[22, 24, 142]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(2, 24)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 48

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(2, 48)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 50

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 142)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 568

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(50, 568)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 618

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[24, 142];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 24, 48], [1, 2, 4, 142, 568]]

new a_3=618;next state=[142, 618];drop ONLY oldest state entry then append new scalar;prefix now=[22, 24, 142, 618]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(2, 142)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 284

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(2, 284)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 286

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 618)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 2472

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(286, 2472)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 2758

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[142, 618];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 142, 284], [1, 3, 4, 618, 2472]]

new a_4=2758;next state=[618, 2758];drop ONLY oldest state entry then append new scalar;prefix now=[22, 24, 142, 618, 2758]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(2, 618)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 1236

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(2, 1236)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 1238

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 2758)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 11032

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(1238, 11032)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 12270

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[618, 2758];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 618, 1236], [1, 4, 4, 2758, 11032]]

new a_5=12270;next state=[2758, 12270];drop ONLY oldest state entry then append new scalar;prefix now=[22, 24, 142, 618, 2758, 12270]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(2, 2758)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 5516

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(2, 5516)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 5518

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 12270)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 49080

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(5518, 49080)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 54598

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[2758, 12270];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 2758, 5516], [1, 5, 4, 12270, 49080]]

new a_6=54598;next state=[12270, 54598];drop ONLY oldest state entry then append new scalar;prefix now=[22, 24, 142, 618, 2758, 12270, 54598]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(2, 12270)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 24540

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(2, 24540)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 24542

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(4, 54598)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 218392

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(24542, 218392)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 242934

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[12270, 54598];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 12270, 24540], [1, 6, 4, 54598, 218392]]

new a_7=242934;next state=[54598, 242934];drop ONLY oldest state entry then append new scalar;prefix now=[22, 24, 142, 618, 2758, 12270, 54598, 242934]

recurrence RIGHT completion:complete ORIGINAL initial state=[22, 24];actual generated prefix=[22, 24, 142, 618, 2758, 12270, 54598, 242934]

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=22;different completions a_1=23 and24;target=7

WHOLE prefixes including initials differ=是;actual targets=[235886, 242934] 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

(22, 23, 24, [22, 23, 138, 600, 2678, 11914, 53014, 235886], [22, 24, 142, 618, 2758, 12270, 54598, 242934], 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。