第 22503 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":0,"initial_values":[2,3,4,5],"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":0,"initial_values":[2,3,4,5],"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=0;ALL initial a_0..a_(r-1)=[2, 3, 4, 5]

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, 2)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 2

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(0, 2)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 2

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(2, 3)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 6

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(2, 6)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 8

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

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 12

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(8, 12)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 20

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(4, 5)

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 20

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(20, 20)

recurrence ORIGINAL n=4 partial sum after coefficient3 = 40

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[2, 3, 4, 5];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 2, 2], [1, 1, 2, 3, 6], [2, 2, 3, 4, 12], [3, 3, 4, 5, 20]]

new a_4=40;next state=[3, 4, 5, 40];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(1, 3)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 3

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(0, 3)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 3

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(2, 4)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 8

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(3, 8)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 11

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

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 15

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(11, 15)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 26

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(4, 40)

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 160

recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(26, 160)

recurrence ORIGINAL n=5 partial sum after coefficient3 = 186

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[3, 4, 5, 40];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 3, 3], [1, 2, 2, 4, 8], [2, 3, 3, 5, 15], [3, 4, 4, 40, 160]]

new a_5=186;next state=[4, 5, 40, 186];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40, 186]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(1, 4)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 4

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(0, 4)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 4

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(2, 5)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 10

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(4, 10)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 14

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(3, 40)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 120

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(14, 120)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 134

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(4, 186)

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 744

recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(134, 744)

recurrence ORIGINAL n=6 partial sum after coefficient3 = 878

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[4, 5, 40, 186];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 4, 4], [1, 3, 2, 5, 10], [2, 4, 3, 40, 120], [3, 5, 4, 186, 744]]

new a_6=878;next state=[5, 40, 186, 878];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40, 186, 878]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(1, 5)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 5

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(0, 5)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 5

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(2, 40)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 80

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(5, 80)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 85

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(3, 186)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 558

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(85, 558)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 643

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(4, 878)

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 3512

recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(643, 3512)

recurrence ORIGINAL n=7 partial sum after coefficient3 = 4155

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[5, 40, 186, 878];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 5, 5], [1, 4, 2, 40, 80], [2, 5, 3, 186, 558], [3, 6, 4, 878, 3512]]

new a_7=4155;next state=[40, 186, 878, 4155];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40, 186, 878, 4155]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 := mul(1, 40)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 = 40

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(0, 40)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 40

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 := mul(2, 186)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 = 372

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(40, 372)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 412

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 := mul(3, 878)

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 = 2634

recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(412, 2634)

recurrence ORIGINAL n=8 partial sum after coefficient2 = 3046

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 := mul(4, 4155)

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 = 16620

recurrence ORIGINAL n=8 partial sum after coefficient3 := sum(3046, 16620)

recurrence ORIGINAL n=8 partial sum after coefficient3 = 19666

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[40, 186, 878, 4155];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 40, 40], [1, 5, 2, 186, 372], [2, 6, 3, 878, 2634], [3, 7, 4, 4155, 16620]]

new a_8=19666;next state=[186, 878, 4155, 19666];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40, 186, 878, 4155, 19666]

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 := mul(1, 186)

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 = 186

recurrence ORIGINAL n=9 partial sum after coefficient0 := sum(0, 186)

recurrence ORIGINAL n=9 partial sum after coefficient0 = 186

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 := mul(2, 878)

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 = 1756

recurrence ORIGINAL n=9 partial sum after coefficient1 := sum(186, 1756)

recurrence ORIGINAL n=9 partial sum after coefficient1 = 1942

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 := mul(3, 4155)

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 = 12465

recurrence ORIGINAL n=9 partial sum after coefficient2 := sum(1942, 12465)

recurrence ORIGINAL n=9 partial sum after coefficient2 = 14407

recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 := mul(4, 19666)

recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 = 78664

recurrence ORIGINAL n=9 partial sum after coefficient3 := sum(14407, 78664)

recurrence ORIGINAL n=9 partial sum after coefficient3 = 93071

recurrence ORIGINAL:ORIGINAL target n=9;previous state a_(n-r)..a_(n-1)=[186, 878, 4155, 19666];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 1, 186, 186], [1, 6, 2, 878, 1756], [2, 7, 3, 4155, 12465], [3, 8, 4, 19666, 78664]]

new a_9=93071;next state=[878, 4155, 19666, 93071];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40, 186, 878, 4155, 19666, 93071]

recurrence ORIGINAL:complete ORIGINAL initial state=[2, 3, 4, 5];actual generated prefix=[2, 3, 4, 5, 40, 186, 878, 4155, 19666, 93071]

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, [2, 3, 4, 5], 9, [2, 3, 4, 5, 40, 186, 878, 4155, 19666, 93071], 93071, 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。