第 22520 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":2,"initial_values":[19,20,21],"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":2,"initial_values":[19,20,21],"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=2;ALL initial a_0..a_(r-1)=[19, 20, 21]

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, 19)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 = 38

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(2, 38)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 40

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 := mul(4, 20)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 = 80

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(40, 80)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 120

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 := mul(6, 21)

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 = 126

recurrence ORIGINAL n=3 partial sum after coefficient2 := sum(120, 126)

recurrence ORIGINAL n=3 partial sum after coefficient2 = 246

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[19, 20, 21];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 19, 38], [1, 1, 4, 20, 80], [2, 2, 6, 21, 126]]

new a_3=246;next state=[20, 21, 246];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 21, 246]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 := mul(2, 20)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 = 40

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(2, 40)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 42

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 := mul(4, 21)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 = 84

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(42, 84)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 126

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 := mul(6, 246)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 = 1476

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(126, 1476)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 1602

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[20, 21, 246];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 20, 40], [1, 2, 4, 21, 84], [2, 3, 6, 246, 1476]]

new a_4=1602;next state=[21, 246, 1602];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 21, 246, 1602]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 := mul(2, 21)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 = 42

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(2, 42)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 44

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 := mul(4, 246)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 = 984

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(44, 984)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 1028

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 := mul(6, 1602)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 = 9612

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(1028, 9612)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 10640

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[21, 246, 1602];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 21, 42], [1, 3, 4, 246, 984], [2, 4, 6, 1602, 9612]]

new a_5=10640;next state=[246, 1602, 10640];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 21, 246, 1602, 10640]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 := mul(2, 246)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 = 492

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(2, 492)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 494

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 := mul(4, 1602)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 = 6408

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(494, 6408)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 6902

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 := mul(6, 10640)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 = 63840

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(6902, 63840)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 70742

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[246, 1602, 10640];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 246, 492], [1, 4, 4, 1602, 6408], [2, 5, 6, 10640, 63840]]

new a_6=70742;next state=[1602, 10640, 70742];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 21, 246, 1602, 10640, 70742]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 := mul(2, 1602)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 = 3204

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(2, 3204)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 3206

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 := mul(4, 10640)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 = 42560

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(3206, 42560)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 45766

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 := mul(6, 70742)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 = 424452

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(45766, 424452)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 470218

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[1602, 10640, 70742];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 1602, 3204], [1, 5, 4, 10640, 42560], [2, 6, 6, 70742, 424452]]

new a_7=470218;next state=[10640, 70742, 470218];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 21, 246, 1602, 10640, 70742, 470218]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_5 := mul(2, 10640)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_5 = 21280

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(2, 21280)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 21282

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_6 := mul(4, 70742)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_6 = 282968

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(21282, 282968)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 304250

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_7 := mul(6, 470218)

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_7 = 2821308

recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(304250, 2821308)

recurrence ORIGINAL n=8 partial sum after coefficient2 = 3125558

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[10640, 70742, 470218];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 10640, 21280], [1, 6, 4, 70742, 282968], [2, 7, 6, 470218, 2821308]]

new a_8=3125558;next state=[70742, 470218, 3125558];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 21, 246, 1602, 10640, 70742, 470218, 3125558]

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_6 := mul(2, 70742)

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_6 = 141484

recurrence ORIGINAL n=9 partial sum after coefficient0 := sum(2, 141484)

recurrence ORIGINAL n=9 partial sum after coefficient0 = 141486

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_7 := mul(4, 470218)

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_7 = 1880872

recurrence ORIGINAL n=9 partial sum after coefficient1 := sum(141486, 1880872)

recurrence ORIGINAL n=9 partial sum after coefficient1 = 2022358

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_8 := mul(6, 3125558)

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_8 = 18753348

recurrence ORIGINAL n=9 partial sum after coefficient2 := sum(2022358, 18753348)

recurrence ORIGINAL n=9 partial sum after coefficient2 = 20775706

recurrence ORIGINAL:ORIGINAL target n=9;previous state a_(n-r)..a_(n-1)=[70742, 470218, 3125558];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 2, 70742, 141484], [1, 7, 4, 470218, 1880872], [2, 8, 6, 3125558, 18753348]]

new a_9=20775706;next state=[470218, 3125558, 20775706];drop ONLY oldest state entry then append new scalar;prefix now=[19, 20, 21, 246, 1602, 10640, 70742, 470218, 3125558, 20775706]

recurrence ORIGINAL:complete ORIGINAL initial state=[19, 20, 21];actual generated prefix=[19, 20, 21, 246, 1602, 10640, 70742, 470218, 3125558, 20775706]

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, [19, 20, 21], 9, [19, 20, 21, 246, 1602, 10640, 70742, 470218, 3125558, 20775706], 20775706, 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。