第 22508 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":[2,4,6],"constant":0,"initial_values":[7,8,9],"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":3,"coefficients":[2,4,6],"constant":0,"initial_values":[7,8,9],"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=3;oldest-to-newest coefficients=[2, 4, 6];constant=0;ALL initial a_0..a_(r-1)=[7, 8, 9]

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(2, 7)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 = 14

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(0, 14)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 14

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 := mul(4, 8)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 = 32

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(14, 32)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 46

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 := mul(6, 9)

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 = 54

recurrence ORIGINAL n=3 partial sum after coefficient2 := sum(46, 54)

recurrence ORIGINAL n=3 partial sum after coefficient2 = 100

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[7, 8, 9];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 7, 14], [1, 1, 4, 8, 32], [2, 2, 6, 9, 54]]

new a_3=100;next state=[8, 9, 100];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 9, 100]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 := mul(2, 8)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 = 16

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(0, 16)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 16

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 := mul(4, 9)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 = 36

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(16, 36)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 52

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 := mul(6, 100)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 = 600

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(52, 600)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 652

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[8, 9, 100];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 8, 16], [1, 2, 4, 9, 36], [2, 3, 6, 100, 600]]

new a_4=652;next state=[9, 100, 652];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 9, 100, 652]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 := mul(2, 9)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 = 18

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(0, 18)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 18

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 := mul(4, 100)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 = 400

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(18, 400)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 418

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 := mul(6, 652)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 = 3912

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(418, 3912)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 4330

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[9, 100, 652];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 9, 18], [1, 3, 4, 100, 400], [2, 4, 6, 652, 3912]]

new a_5=4330;next state=[100, 652, 4330];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 9, 100, 652, 4330]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 := mul(2, 100)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 = 200

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(0, 200)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 200

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 := mul(4, 652)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 = 2608

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(200, 2608)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 2808

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 := mul(6, 4330)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 = 25980

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(2808, 25980)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 28788

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[100, 652, 4330];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 100, 200], [1, 4, 4, 652, 2608], [2, 5, 6, 4330, 25980]]

new a_6=28788;next state=[652, 4330, 28788];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 9, 100, 652, 4330, 28788]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 := mul(2, 652)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 = 1304

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(0, 1304)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 1304

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 := mul(4, 4330)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 = 17320

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(1304, 17320)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 18624

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 := mul(6, 28788)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 = 172728

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(18624, 172728)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 191352

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[652, 4330, 28788];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 652, 1304], [1, 5, 4, 4330, 17320], [2, 6, 6, 28788, 172728]]

new a_7=191352;next state=[4330, 28788, 191352];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 9, 100, 652, 4330, 28788, 191352]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_5 := mul(2, 4330)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_5 = 8660

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(0, 8660)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 8660

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_6 := mul(4, 28788)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_6 = 115152

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(8660, 115152)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 123812

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_7 := mul(6, 191352)

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_7 = 1148112

recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(123812, 1148112)

recurrence ORIGINAL n=8 partial sum after coefficient2 = 1271924

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[4330, 28788, 191352];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 4330, 8660], [1, 6, 4, 28788, 115152], [2, 7, 6, 191352, 1148112]]

new a_8=1271924;next state=[28788, 191352, 1271924];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 9, 100, 652, 4330, 28788, 191352, 1271924]

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_6 := mul(2, 28788)

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_6 = 57576

recurrence ORIGINAL n=9 partial sum after coefficient0 := sum(0, 57576)

recurrence ORIGINAL n=9 partial sum after coefficient0 = 57576

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_7 := mul(4, 191352)

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_7 = 765408

recurrence ORIGINAL n=9 partial sum after coefficient1 := sum(57576, 765408)

recurrence ORIGINAL n=9 partial sum after coefficient1 = 822984

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_8 := mul(6, 1271924)

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_8 = 7631544

recurrence ORIGINAL n=9 partial sum after coefficient2 := sum(822984, 7631544)

recurrence ORIGINAL n=9 partial sum after coefficient2 = 8454528

recurrence ORIGINAL:ORIGINAL target n=9;previous state a_(n-r)..a_(n-1)=[28788, 191352, 1271924];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 2, 28788, 57576], [1, 7, 4, 191352, 765408], [2, 8, 6, 1271924, 7631544]]

new a_9=8454528;next state=[191352, 1271924, 8454528];drop ONLY oldest state entry then append new scalar;prefix now=[7, 8, 9, 100, 652, 4330, 28788, 191352, 1271924, 8454528]

recurrence ORIGINAL:complete ORIGINAL initial state=[7, 8, 9];actual generated prefix=[7, 8, 9, 100, 652, 4330, 28788, 191352, 1271924, 8454528]

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

• S052 · 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。
• S051 · 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。
• S050 · 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。
• S049 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_8，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 partial sum after coefficient1，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_7，不以小數近似替代。
• S046 · 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。
• S045 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_6，不以小數近似替代。
• 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, [7, 8, 9], 9, [7, 8, 9, 100, 652, 4330, 28788, 191352, 1271924, 8454528], 8454528, 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。