第 22521 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-forward-generation；資料={"order":4,"coefficients":[1,2,3,4],"constant":-2,"initial_values":[20,21,22,23],"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-forward-generation；資料={"order":4,"coefficients":[1,2,3,4],"constant":-2,"initial_values":[20,21,22,23],"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-forward-generation;original snapshot and mathematics retained

explicit ordered source output fields=[order, initial_values, target_n, prefix, target_value, certificate]

recurrence ORIGINAL:ORIGINAL order=4;oldest-to-newest coefficients=[1, 2, 3, 4];constant=-2;ALL initial a_0..a_(r-1)=[20, 21, 22, 23]

for EVERY n>=4,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=4;forward rule is total and single-valued

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 := mul(1, 20)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 20

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(-2, 20)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 18

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(2, 21)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 42

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(18, 42)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 60

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(3, 22)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 66

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(60, 66)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 126

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(4, 23)

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 92

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(126, 92)

recurrence ORIGINAL n=4 partial sum after coefficient3 = 218

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[20, 21, 22, 23];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 20, 20], [1, 1, 2, 21, 42], [2, 2, 3, 22, 66], [3, 3, 4, 23, 92]]

new a_4=218;next state=[21, 22, 23, 218];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 22, 23, 218]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(1, 21)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 21

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(-2, 21)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 19

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(2, 22)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 44

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(19, 44)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 63

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(3, 23)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 69

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(63, 69)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 132

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(4, 218)

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 872

recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(132, 872)

recurrence ORIGINAL n=5 partial sum after coefficient3 = 1004

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[21, 22, 23, 218];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 21, 21], [1, 2, 2, 22, 44], [2, 3, 3, 23, 69], [3, 4, 4, 218, 872]]

new a_5=1004;next state=[22, 23, 218, 1004];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 22, 23, 218, 1004]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(1, 22)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 22

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(-2, 22)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 20

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(2, 23)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 46

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(20, 46)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 66

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(3, 218)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 654

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(66, 654)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 720

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(4, 1004)

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 4016

recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(720, 4016)

recurrence ORIGINAL n=6 partial sum after coefficient3 = 4736

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[22, 23, 218, 1004];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 22, 22], [1, 3, 2, 23, 46], [2, 4, 3, 218, 654], [3, 5, 4, 1004, 4016]]

new a_6=4736;next state=[23, 218, 1004, 4736];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 22, 23, 218, 1004, 4736]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(1, 23)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 23

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(-2, 23)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 21

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(2, 218)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 436

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(21, 436)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 457

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(3, 1004)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 3012

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(457, 3012)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 3469

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(4, 4736)

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 18944

recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(3469, 18944)

recurrence ORIGINAL n=7 partial sum after coefficient3 = 22413

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[23, 218, 1004, 4736];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 23, 23], [1, 4, 2, 218, 436], [2, 5, 3, 1004, 3012], [3, 6, 4, 4736, 18944]]

new a_7=22413;next state=[218, 1004, 4736, 22413];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 22, 23, 218, 1004, 4736, 22413]

recurrence ORIGINAL:complete ORIGINAL initial state=[20, 21, 22, 23];actual generated prefix=[20, 21, 22, 23, 218, 1004, 4736, 22413]

ANY two sequences with SAME r initials and SAME total rule agree at n=4 by substitution,then at every later n by induction;this determines the requested prefix only

逐步反向核對

• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S037 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 partial sum after coefficient3，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 partial sum after coefficient2，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 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 ORIGINAL n=7 ORIGINAL coefficient1 times a_4，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 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 ORIGINAL n=7 ORIGINAL coefficient0 times a_3，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 — 以精確有理數重算 recurrence ORIGINAL n=6 partial sum after coefficient3，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 partial sum after coefficient2，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 partial sum after coefficient1，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 partial sum after coefficient0，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2，不以小數近似替代。
• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 partial sum after coefficient3，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 partial sum after coefficient2，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 partial sum after coefficient1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 partial sum after coefficient0，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1，不以小數近似替代。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 partial sum after coefficient3，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 partial sum after coefficient2，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 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 ORIGINAL n=4 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 ORIGINAL n=4 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 ORIGINAL n=4 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, [20, 21, 22, 23], 7, [20, 21, 22, 23, 218, 1004, 4736, 22413], 22413, exact_forward_application_only)

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。