第 22572 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":[1,3],"constant":1,"first_initial":21,"second_left":22,"second_right":23,"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":[1,3],"constant":1,"first_initial":21,"second_left":22,"second_right":23,"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, 3];constant=1;ALL initial a_0..a_(r-1)=[21, 22]

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, 21)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 21

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(1, 21)

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

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 22)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 66

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(22, 66)

recurrence LEFT completion n=2 partial sum after coefficient1 = 88

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[21, 22];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 21, 21], [1, 1, 3, 22, 66]]

new a_2=88;next state=[22, 88];drop ONLY oldest state entry then append new scalar;prefix now=[21, 22, 88]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 22)

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

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(1, 22)

recurrence LEFT completion n=3 partial sum after coefficient0 = 23

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 88)

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

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(23, 264)

recurrence LEFT completion n=3 partial sum after coefficient1 = 287

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[22, 88];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 22, 22], [1, 2, 3, 88, 264]]

new a_3=287;next state=[88, 287];drop ONLY oldest state entry then append new scalar;prefix now=[21, 22, 88, 287]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 88)

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

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(1, 88)

recurrence LEFT completion n=4 partial sum after coefficient0 = 89

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 287)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 861

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(89, 861)

recurrence LEFT completion n=4 partial sum after coefficient1 = 950

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[88, 287];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 88, 88], [1, 3, 3, 287, 861]]

new a_4=950;next state=[287, 950];drop ONLY oldest state entry then append new scalar;prefix now=[21, 22, 88, 287, 950]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 287)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 287

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(1, 287)

recurrence LEFT completion n=5 partial sum after coefficient0 = 288

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 950)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 2850

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(288, 2850)

recurrence LEFT completion n=5 partial sum after coefficient1 = 3138

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[287, 950];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 287, 287], [1, 4, 3, 950, 2850]]

new a_5=3138;next state=[950, 3138];drop ONLY oldest state entry then append new scalar;prefix now=[21, 22, 88, 287, 950, 3138]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 950)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 950

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(1, 950)

recurrence LEFT completion n=6 partial sum after coefficient0 = 951

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 3138)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 9414

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(951, 9414)

recurrence LEFT completion n=6 partial sum after coefficient1 = 10365

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[950, 3138];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 950, 950], [1, 5, 3, 3138, 9414]]

new a_6=10365;next state=[3138, 10365];drop ONLY oldest state entry then append new scalar;prefix now=[21, 22, 88, 287, 950, 3138, 10365]

recurrence LEFT completion:complete ORIGINAL initial state=[21, 22];actual generated prefix=[21, 22, 88, 287, 950, 3138, 10365]

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, 3];constant=1;ALL initial a_0..a_(r-1)=[21, 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 RIGHT completion n=2 ORIGINAL coefficient0 times a_0 := mul(1, 21)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 21

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(1, 21)

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

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 23)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 69

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(22, 69)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 91

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[21, 23];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 21, 21], [1, 1, 3, 23, 69]]

new a_2=91;next state=[23, 91];drop ONLY oldest state entry then append new scalar;prefix now=[21, 23, 91]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 23)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 23

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(1, 23)

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

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 91)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 273

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(24, 273)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 297

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[23, 91];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 23, 23], [1, 2, 3, 91, 273]]

new a_3=297;next state=[91, 297];drop ONLY oldest state entry then append new scalar;prefix now=[21, 23, 91, 297]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 91)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 91

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(1, 91)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 92

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 297)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 891

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(92, 891)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 983

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[91, 297];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 91, 91], [1, 3, 3, 297, 891]]

new a_4=983;next state=[297, 983];drop ONLY oldest state entry then append new scalar;prefix now=[21, 23, 91, 297, 983]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 297)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 297

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(1, 297)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 298

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 983)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 2949

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(298, 2949)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 3247

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[297, 983];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 297, 297], [1, 4, 3, 983, 2949]]

new a_5=3247;next state=[983, 3247];drop ONLY oldest state entry then append new scalar;prefix now=[21, 23, 91, 297, 983, 3247]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 983)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 983

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(1, 983)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 984

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 3247)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 9741

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(984, 9741)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 10725

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[983, 3247];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 983, 983], [1, 5, 3, 3247, 9741]]

new a_6=10725;next state=[3247, 10725];drop ONLY oldest state entry then append new scalar;prefix now=[21, 23, 91, 297, 983, 3247, 10725]

recurrence RIGHT completion:complete ORIGINAL initial state=[21, 23];actual generated prefix=[21, 23, 91, 297, 983, 3247, 10725]

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=21;different completions a_1=22 and23;target=6

WHOLE prefixes including initials differ=是;actual targets=[10365, 10725] 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

(21, 22, 23, [21, 22, 88, 287, 950, 3138, 10365], [21, 23, 91, 297, 983, 3247, 10725], 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。