第 22569 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":[1,5],"constant":2,"first_initial":18,"second_left":19,"second_right":20,"target_n":6}。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":[1,5],"constant":2,"first_initial":18,"second_left":19,"second_right":20,"target_n":6}。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=[1, 5];constant=2;ALL initial a_0..a_(r-1)=[18, 19]

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(1, 18)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 18

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(2, 18)

recurrence LEFT completion n=2 partial sum after coefficient0 = 20

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(5, 19)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 95

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(20, 95)

recurrence LEFT completion n=2 partial sum after coefficient1 = 115

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[18, 19];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 18, 18], [1, 1, 5, 19, 95]]

new a_2=115;next state=[19, 115];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 115]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 19)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 19

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(2, 19)

recurrence LEFT completion n=3 partial sum after coefficient0 = 21

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(5, 115)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 575

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(21, 575)

recurrence LEFT completion n=3 partial sum after coefficient1 = 596

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[19, 115];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 19, 19], [1, 2, 5, 115, 575]]

new a_3=596;next state=[115, 596];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 115, 596]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 115)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 115

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(2, 115)

recurrence LEFT completion n=4 partial sum after coefficient0 = 117

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(5, 596)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 2980

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(117, 2980)

recurrence LEFT completion n=4 partial sum after coefficient1 = 3097

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[115, 596];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 115, 115], [1, 3, 5, 596, 2980]]

new a_4=3097;next state=[596, 3097];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 115, 596, 3097]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 596)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 596

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(2, 596)

recurrence LEFT completion n=5 partial sum after coefficient0 = 598

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(5, 3097)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 15485

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(598, 15485)

recurrence LEFT completion n=5 partial sum after coefficient1 = 16083

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[596, 3097];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 596, 596], [1, 4, 5, 3097, 15485]]

new a_5=16083;next state=[3097, 16083];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 115, 596, 3097, 16083]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 3097)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 3097

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(2, 3097)

recurrence LEFT completion n=6 partial sum after coefficient0 = 3099

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(5, 16083)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 80415

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(3099, 80415)

recurrence LEFT completion n=6 partial sum after coefficient1 = 83514

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[3097, 16083];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 3097, 3097], [1, 5, 5, 16083, 80415]]

new a_6=83514;next state=[16083, 83514];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 115, 596, 3097, 16083, 83514]

recurrence LEFT completion:complete ORIGINAL initial state=[18, 19];actual generated prefix=[18, 19, 115, 596, 3097, 16083, 83514]

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=[1, 5];constant=2;ALL initial a_0..a_(r-1)=[18, 20]

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(1, 18)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 18

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(2, 18)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 20

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(5, 20)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 100

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(20, 100)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 120

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[18, 20];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 18, 18], [1, 1, 5, 20, 100]]

new a_2=120;next state=[20, 120];drop ONLY oldest state entry then append new scalar;prefix now=[18, 20, 120]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 20)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 20

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(2, 20)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 22

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(5, 120)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 600

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(22, 600)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 622

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[20, 120];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 20, 20], [1, 2, 5, 120, 600]]

new a_3=622;next state=[120, 622];drop ONLY oldest state entry then append new scalar;prefix now=[18, 20, 120, 622]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 120)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 120

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(2, 120)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 122

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(5, 622)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 3110

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(122, 3110)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 3232

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[120, 622];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 120, 120], [1, 3, 5, 622, 3110]]

new a_4=3232;next state=[622, 3232];drop ONLY oldest state entry then append new scalar;prefix now=[18, 20, 120, 622, 3232]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 622)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 622

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(2, 622)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 624

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(5, 3232)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 16160

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(624, 16160)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 16784

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[622, 3232];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 622, 622], [1, 4, 5, 3232, 16160]]

new a_5=16784;next state=[3232, 16784];drop ONLY oldest state entry then append new scalar;prefix now=[18, 20, 120, 622, 3232, 16784]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 3232)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 3232

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(2, 3232)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 3234

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(5, 16784)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 83920

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(3234, 83920)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 87154

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[3232, 16784];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 3232, 3232], [1, 5, 5, 16784, 83920]]

new a_6=87154;next state=[16784, 87154];drop ONLY oldest state entry then append new scalar;prefix now=[18, 20, 120, 622, 3232, 16784, 87154]

recurrence RIGHT completion:complete ORIGINAL initial state=[18, 20];actual generated prefix=[18, 20, 120, 622, 3232, 16784, 87154]

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=18;different completions a_1=19 and20;target=6

WHOLE prefixes including initials differ=是;actual targets=[83514, 87154] MAY coincide;one_initial_uniqueness_claim fixed No

逐步反向核對

• S056 · 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。
• S055 · 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。
• 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=6 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=6 ORIGINAL coefficient1 times a_5，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=6 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=6 ORIGINAL coefficient0 times a_4，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=5 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=5 ORIGINAL coefficient1 times a_4，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=5 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=5 ORIGINAL coefficient0 times a_3，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=4 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=4 ORIGINAL coefficient1 times a_3，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=4 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=4 ORIGINAL coefficient0 times a_2，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=3 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=3 ORIGINAL coefficient1 times a_2，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=3 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=3 ORIGINAL coefficient0 times a_1，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 — 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。
• S032 · 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。
• S031 · 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。
• S030 · 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。
• S029 · 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。
• S028 · 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。
• 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

(18, 19, 20, [18, 19, 115, 596, 3097, 16083, 83514], [18, 20, 120, 622, 3232, 16784, 87154], 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。