第 22515 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":[14,15,16,17],"target_n":9}。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":[14,15,16,17],"target_n":9}。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)=[14, 15, 16, 17]

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, 14)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 14

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(2, 14)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 16

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(2, 15)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 30

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(16, 30)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 46

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(3, 16)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 48

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(46, 48)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 94

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(4, 17)

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 68

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(94, 68)

recurrence ORIGINAL n=4 partial sum after coefficient3 = 162

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[14, 15, 16, 17];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 14, 14], [1, 1, 2, 15, 30], [2, 2, 3, 16, 48], [3, 3, 4, 17, 68]]

new a_4=162;next state=[15, 16, 17, 162];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(1, 15)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 15

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(2, 15)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 17

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(2, 16)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 32

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(17, 32)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 49

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(3, 17)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 51

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(49, 51)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 100

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(4, 162)

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 648

recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(100, 648)

recurrence ORIGINAL n=5 partial sum after coefficient3 = 748

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[15, 16, 17, 162];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 15, 15], [1, 2, 2, 16, 32], [2, 3, 3, 17, 51], [3, 4, 4, 162, 648]]

new a_5=748;next state=[16, 17, 162, 748];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(1, 16)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 16

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(2, 16)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 18

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(2, 17)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 34

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(18, 34)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 52

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(3, 162)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 486

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(52, 486)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 538

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(4, 748)

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 2992

recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(538, 2992)

recurrence ORIGINAL n=6 partial sum after coefficient3 = 3530

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[16, 17, 162, 748];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 16, 16], [1, 3, 2, 17, 34], [2, 4, 3, 162, 486], [3, 5, 4, 748, 2992]]

new a_6=3530;next state=[17, 162, 748, 3530];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(1, 17)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 17

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(2, 17)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 19

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(2, 162)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 324

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(19, 324)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 343

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(3, 748)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 2244

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(343, 2244)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 2587

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(4, 3530)

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 14120

recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(2587, 14120)

recurrence ORIGINAL n=7 partial sum after coefficient3 = 16707

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[17, 162, 748, 3530];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 17, 17], [1, 4, 2, 162, 324], [2, 5, 3, 748, 2244], [3, 6, 4, 3530, 14120]]

new a_7=16707;next state=[162, 748, 3530, 16707];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530, 16707]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 := mul(1, 162)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 = 162

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(2, 162)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 164

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 := mul(2, 748)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 = 1496

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(164, 1496)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 1660

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 := mul(3, 3530)

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 = 10590

recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(1660, 10590)

recurrence ORIGINAL n=8 partial sum after coefficient2 = 12250

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 := mul(4, 16707)

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 = 66828

recurrence ORIGINAL n=8 partial sum after coefficient3 := sum(12250, 66828)

recurrence ORIGINAL n=8 partial sum after coefficient3 = 79078

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[162, 748, 3530, 16707];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 162, 162], [1, 5, 2, 748, 1496], [2, 6, 3, 3530, 10590], [3, 7, 4, 16707, 66828]]

new a_8=79078;next state=[748, 3530, 16707, 79078];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530, 16707, 79078]

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 := mul(1, 748)

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 = 748

recurrence ORIGINAL n=9 partial sum after coefficient0 := sum(2, 748)

recurrence ORIGINAL n=9 partial sum after coefficient0 = 750

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 := mul(2, 3530)

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 = 7060

recurrence ORIGINAL n=9 partial sum after coefficient1 := sum(750, 7060)

recurrence ORIGINAL n=9 partial sum after coefficient1 = 7810

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 := mul(3, 16707)

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 = 50121

recurrence ORIGINAL n=9 partial sum after coefficient2 := sum(7810, 50121)

recurrence ORIGINAL n=9 partial sum after coefficient2 = 57931

recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 := mul(4, 79078)

recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 = 316312

recurrence ORIGINAL n=9 partial sum after coefficient3 := sum(57931, 316312)

recurrence ORIGINAL n=9 partial sum after coefficient3 = 374243

recurrence ORIGINAL:ORIGINAL target n=9;previous state a_(n-r)..a_(n-1)=[748, 3530, 16707, 79078];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 1, 748, 748], [1, 6, 2, 3530, 7060], [2, 7, 3, 16707, 50121], [3, 8, 4, 79078, 316312]]

new a_9=374243;next state=[3530, 16707, 79078, 374243];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243]

recurrence ORIGINAL:complete ORIGINAL initial state=[14, 15, 16, 17];actual generated prefix=[14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243]

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

• S057 · 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。
• 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 — 以精確有理數重算 recurrence ORIGINAL n=9 partial sum after coefficient3，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 partial sum after coefficient2，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 partial sum after coefficient1，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 partial sum after coefficient0，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5，不以小數近似替代。
• S047 · 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。
• S046 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 partial sum after coefficient3，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7，不以小數近似替代。
• S044 · 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。
• S043 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6，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 partial sum after coefficient1，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 ORIGINAL coefficient1 times a_5，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 partial sum after coefficient0，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 ORIGINAL coefficient0 times a_4，不以小數近似替代。
• 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, [14, 15, 16, 17], 9, [14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243], 374243, 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。