第 22506 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":[2,4,6,8],"constant":-2,"initial_values":[5,6,7,8],"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":4,"coefficients":[2,4,6,8],"constant":-2,"initial_values":[5,6,7,8],"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=4;oldest-to-newest coefficients=[2, 4, 6, 8];constant=-2;ALL initial a_0..a_(r-1)=[5, 6, 7, 8]

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(2, 5)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 10

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(-2, 10)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 8

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(4, 6)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 24

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(8, 24)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 32

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(6, 7)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 42

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(32, 42)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 74

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(8, 8)

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 64

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(74, 64)

recurrence ORIGINAL n=4 partial sum after coefficient3 = 138

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[5, 6, 7, 8];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 5, 10], [1, 1, 4, 6, 24], [2, 2, 6, 7, 42], [3, 3, 8, 8, 64]]

new a_4=138;next state=[6, 7, 8, 138];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 7, 8, 138]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(2, 6)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 12

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(-2, 12)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 10

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(4, 7)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 28

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(10, 28)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 38

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(6, 8)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 48

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(38, 48)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 86

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(8, 138)

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 1104

recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(86, 1104)

recurrence ORIGINAL n=5 partial sum after coefficient3 = 1190

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[6, 7, 8, 138];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 6, 12], [1, 2, 4, 7, 28], [2, 3, 6, 8, 48], [3, 4, 8, 138, 1104]]

new a_5=1190;next state=[7, 8, 138, 1190];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 7, 8, 138, 1190]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(2, 7)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 14

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(-2, 14)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 12

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(4, 8)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 32

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(12, 32)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 44

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(6, 138)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 828

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(44, 828)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 872

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(8, 1190)

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 9520

recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(872, 9520)

recurrence ORIGINAL n=6 partial sum after coefficient3 = 10392

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[7, 8, 138, 1190];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 7, 14], [1, 3, 4, 8, 32], [2, 4, 6, 138, 828], [3, 5, 8, 1190, 9520]]

new a_6=10392;next state=[8, 138, 1190, 10392];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 7, 8, 138, 1190, 10392]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(2, 8)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 16

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(-2, 16)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 14

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(4, 138)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 552

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(14, 552)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 566

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(6, 1190)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 7140

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(566, 7140)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 7706

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(8, 10392)

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 83136

recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(7706, 83136)

recurrence ORIGINAL n=7 partial sum after coefficient3 = 90842

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[8, 138, 1190, 10392];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 8, 16], [1, 4, 4, 138, 552], [2, 5, 6, 1190, 7140], [3, 6, 8, 10392, 83136]]

new a_7=90842;next state=[138, 1190, 10392, 90842];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 7, 8, 138, 1190, 10392, 90842]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 := mul(2, 138)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 = 276

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(-2, 276)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 274

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 := mul(4, 1190)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 = 4760

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(274, 4760)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 5034

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 := mul(6, 10392)

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 = 62352

recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(5034, 62352)

recurrence ORIGINAL n=8 partial sum after coefficient2 = 67386

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 := mul(8, 90842)

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 = 726736

recurrence ORIGINAL n=8 partial sum after coefficient3 := sum(67386, 726736)

recurrence ORIGINAL n=8 partial sum after coefficient3 = 794122

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[138, 1190, 10392, 90842];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 138, 276], [1, 5, 4, 1190, 4760], [2, 6, 6, 10392, 62352], [3, 7, 8, 90842, 726736]]

new a_8=794122;next state=[1190, 10392, 90842, 794122];drop ONLY oldest state entry then append new scalar;prefix now=[5, 6, 7, 8, 138, 1190, 10392, 90842, 794122]

recurrence ORIGINAL:complete ORIGINAL initial state=[5, 6, 7, 8];actual generated prefix=[5, 6, 7, 8, 138, 1190, 10392, 90842, 794122]

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

• S048 · 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。
• 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, [5, 6, 7, 8], 8, [5, 6, 7, 8, 138, 1190, 10392, 90842, 794122], 794122, 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。