第 22523 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":3,"coefficients":[1,2,3],"constant":0,"initial_values":[22,23,24],"target_n":8}。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":3,"coefficients":[1,2,3],"constant":0,"initial_values":[22,23,24],"target_n":8}。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=3;oldest-to-newest coefficients=[1, 2, 3];constant=0;ALL initial a_0..a_(r-1)=[22, 23, 24]

for EVERY n>=3,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=3;forward rule is total and single-valued

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 := mul(1, 22)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 = 22

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(0, 22)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 22

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

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 = 46

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(22, 46)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 68

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 := mul(3, 24)

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 = 72

recurrence ORIGINAL n=3 partial sum after coefficient2 := sum(68, 72)

recurrence ORIGINAL n=3 partial sum after coefficient2 = 140

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[22, 23, 24];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 22, 22], [1, 1, 2, 23, 46], [2, 2, 3, 24, 72]]

new a_3=140;next state=[23, 24, 140];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 24, 140]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 := mul(1, 23)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 = 23

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(0, 23)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 23

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 := mul(2, 24)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 = 48

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(23, 48)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 71

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 := mul(3, 140)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 = 420

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(71, 420)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 491

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[23, 24, 140];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 23, 23], [1, 2, 2, 24, 48], [2, 3, 3, 140, 420]]

new a_4=491;next state=[24, 140, 491];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 24, 140, 491]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 := mul(1, 24)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 = 24

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(0, 24)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 24

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 := mul(2, 140)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 = 280

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(24, 280)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 304

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 := mul(3, 491)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 = 1473

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(304, 1473)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 1777

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[24, 140, 491];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 24, 24], [1, 3, 2, 140, 280], [2, 4, 3, 491, 1473]]

new a_5=1777;next state=[140, 491, 1777];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 24, 140, 491, 1777]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 := mul(1, 140)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 = 140

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(0, 140)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 140

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 := mul(2, 491)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 = 982

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(140, 982)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 1122

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 := mul(3, 1777)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 = 5331

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(1122, 5331)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 6453

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[140, 491, 1777];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 140, 140], [1, 4, 2, 491, 982], [2, 5, 3, 1777, 5331]]

new a_6=6453;next state=[491, 1777, 6453];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 24, 140, 491, 1777, 6453]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 := mul(1, 491)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 = 491

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(0, 491)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 491

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 := mul(2, 1777)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 = 3554

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(491, 3554)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 4045

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 := mul(3, 6453)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 = 19359

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(4045, 19359)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 23404

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[491, 1777, 6453];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 491, 491], [1, 5, 2, 1777, 3554], [2, 6, 3, 6453, 19359]]

new a_7=23404;next state=[1777, 6453, 23404];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 24, 140, 491, 1777, 6453, 23404]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_5 := mul(1, 1777)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_5 = 1777

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(0, 1777)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 1777

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_6 := mul(2, 6453)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_6 = 12906

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(1777, 12906)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 14683

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_7 := mul(3, 23404)

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_7 = 70212

recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(14683, 70212)

recurrence ORIGINAL n=8 partial sum after coefficient2 = 84895

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[1777, 6453, 23404];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 1, 1777, 1777], [1, 6, 2, 6453, 12906], [2, 7, 3, 23404, 70212]]

new a_8=84895;next state=[6453, 23404, 84895];drop ONLY oldest state entry then append new scalar;prefix now=[22, 23, 24, 140, 491, 1777, 6453, 23404, 84895]

recurrence ORIGINAL:complete ORIGINAL initial state=[22, 23, 24];actual generated prefix=[22, 23, 24, 140, 491, 1777, 6453, 23404, 84895]

ANY two sequences with SAME r initials and SAME total rule agree at n=3 by substitution,then at every later n by induction;this determines the requested prefix only

逐步反向核對

• S045 · 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。
• 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 ORIGINAL n=8 partial sum after coefficient2，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_7，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 partial sum after coefficient1，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_6，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 partial sum after coefficient0，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_5，不以小數近似替代。
• S037 · 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。
• S036 · 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。
• S035 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6，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 partial sum after coefficient1，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 ORIGINAL coefficient1 times a_5，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 partial sum after coefficient0，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 ORIGINAL coefficient0 times a_4，不以小數近似替代。
• S030 · 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。
• S029 · 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。
• S028 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5，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 partial sum after coefficient1，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 ORIGINAL coefficient1 times a_4，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 partial sum after coefficient0，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 ORIGINAL coefficient0 times a_3，不以小數近似替代。
• S023 · 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。
• S022 · 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。
• S021 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4，不以小數近似替代。
• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3，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 partial sum after coefficient0，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 ORIGINAL coefficient0 times a_2，不以小數近似替代。
• S016 · 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。
• S015 · 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。
• S014 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3，不以小數近似替代。
• S013 · 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。
• S012 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2，不以小數近似替代。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1，不以小數近似替代。
• S009 · 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。
• S008 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=3 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=3 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=3 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=3 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=3 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=3 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

(3, [22, 23, 24], 8, [22, 23, 24, 140, 491, 1777, 6453, 23404, 84895], 84895, 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。