第 22511 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":[1,2,3],"constant":-2,"initial_values":[10,11,12],"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":3,"coefficients":[1,2,3],"constant":-2,"initial_values":[10,11,12],"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=3;oldest-to-newest coefficients=[1, 2, 3];constant=-2;ALL initial a_0..a_(r-1)=[10, 11, 12]

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(1, 10)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 = 10

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(-2, 10)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 8

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 := mul(2, 11)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 = 22

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(8, 22)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 30

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 := mul(3, 12)

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 = 36

recurrence ORIGINAL n=3 partial sum after coefficient2 := sum(30, 36)

recurrence ORIGINAL n=3 partial sum after coefficient2 = 66

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[10, 11, 12];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 10, 10], [1, 1, 2, 11, 22], [2, 2, 3, 12, 36]]

new a_3=66;next state=[11, 12, 66];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 12, 66]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 := mul(1, 11)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 = 11

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(-2, 11)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 9

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 := mul(2, 12)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 = 24

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(9, 24)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 33

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 := mul(3, 66)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 = 198

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(33, 198)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 231

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[11, 12, 66];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 11, 11], [1, 2, 2, 12, 24], [2, 3, 3, 66, 198]]

new a_4=231;next state=[12, 66, 231];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 12, 66, 231]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 := mul(1, 12)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 = 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_3 := mul(2, 66)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 = 132

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(10, 132)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 142

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 := mul(3, 231)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 = 693

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(142, 693)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 835

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[12, 66, 231];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 12, 12], [1, 3, 2, 66, 132], [2, 4, 3, 231, 693]]

new a_5=835;next state=[66, 231, 835];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 12, 66, 231, 835]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 := mul(1, 66)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 = 66

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(-2, 66)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 64

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 := mul(2, 231)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 = 462

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(64, 462)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 526

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 := mul(3, 835)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 = 2505

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(526, 2505)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 3031

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[66, 231, 835];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 66, 66], [1, 4, 2, 231, 462], [2, 5, 3, 835, 2505]]

new a_6=3031;next state=[231, 835, 3031];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 12, 66, 231, 835, 3031]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 := mul(1, 231)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 = 231

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(-2, 231)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 229

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 := mul(2, 835)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 = 1670

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(229, 1670)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 1899

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 := mul(3, 3031)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 = 9093

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(1899, 9093)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 10992

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[231, 835, 3031];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 231, 231], [1, 5, 2, 835, 1670], [2, 6, 3, 3031, 9093]]

new a_7=10992;next state=[835, 3031, 10992];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 12, 66, 231, 835, 3031, 10992]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_5 := mul(1, 835)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_5 = 835

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(-2, 835)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 833

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_6 := mul(2, 3031)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_6 = 6062

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(833, 6062)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 6895

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_7 := mul(3, 10992)

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_7 = 32976

recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(6895, 32976)

recurrence ORIGINAL n=8 partial sum after coefficient2 = 39871

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[835, 3031, 10992];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 1, 835, 835], [1, 6, 2, 3031, 6062], [2, 7, 3, 10992, 32976]]

new a_8=39871;next state=[3031, 10992, 39871];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 12, 66, 231, 835, 3031, 10992, 39871]

recurrence ORIGINAL:complete ORIGINAL initial state=[10, 11, 12];actual generated prefix=[10, 11, 12, 66, 231, 835, 3031, 10992, 39871]

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

• S045 · 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。
• 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, [10, 11, 12], 8, [10, 11, 12, 66, 231, 835, 3031, 10992, 39871], 39871, 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。