第 22535 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-initial-value-uniqueness；資料={"order":2,"coefficients":[2,4],"constant":2,"initial_values":[9,10],"target_n":6}。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-initial-value-uniqueness；資料={"order":2,"coefficients":[2,4],"constant":2,"initial_values":[9,10],"target_n":6}。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-initial-value-uniqueness;original snapshot and mathematics retained

explicit ordered source output fields=[order, initial_count, target_n, target_value, unique_prefix, certificate]

recurrence ORIGINAL:ORIGINAL order=2;oldest-to-newest coefficients=[2, 4];constant=2;ALL initial a_0..a_(r-1)=[9, 10]

for EVERY n>=2,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=2;forward rule is total and single-valued

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 := mul(2, 9)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 18

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(2, 18)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 20

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(4, 10)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 40

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(20, 40)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 60

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[9, 10];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 9, 18], [1, 1, 4, 10, 40]]

new a_2=60;next state=[10, 60];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 60]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(2, 10)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 20

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(2, 20)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 22

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(4, 60)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 240

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(22, 240)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 262

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[10, 60];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 10, 20], [1, 2, 4, 60, 240]]

new a_3=262;next state=[60, 262];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 60, 262]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(2, 60)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 120

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(2, 120)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 122

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(4, 262)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 1048

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(122, 1048)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 1170

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[60, 262];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 60, 120], [1, 3, 4, 262, 1048]]

new a_4=1170;next state=[262, 1170];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 60, 262, 1170]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(2, 262)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 524

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(2, 524)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 526

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(4, 1170)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 4680

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(526, 4680)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 5206

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[262, 1170];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 262, 524], [1, 4, 4, 1170, 4680]]

new a_5=5206;next state=[1170, 5206];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 60, 262, 1170, 5206]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 := mul(2, 1170)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 = 2340

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(2, 2340)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 2342

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 := mul(4, 5206)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 = 20824

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(2342, 20824)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 23166

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1170, 5206];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 1170, 2340], [1, 5, 4, 5206, 20824]]

new a_6=23166;next state=[5206, 23166];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 60, 262, 1170, 5206, 23166]

recurrence ORIGINAL:complete ORIGINAL initial state=[9, 10];actual generated prefix=[9, 10, 60, 262, 1170, 5206, 23166]

ANY two sequences with SAME r initials and SAME total rule agree at n=2 by substitution,then at every later n by induction;this determines the requested prefix only

逐步反向核對

• S028 · 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。
• S027 · 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。
• S026 · 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。
• S025 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5，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 coefficient0，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 coefficient0 times a_4，不以小數近似替代。
• S022 · 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。
• S021 · 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。
• S020 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4，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 partial sum after coefficient0，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 coefficient0 times a_3，不以小數近似替代。
• S017 · 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。
• S016 · 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。
• S015 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3，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 partial sum after coefficient0，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 ORIGINAL coefficient0 times a_2，不以小數近似替代。
• S012 · 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。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2，不以小數近似替代。
• S009 · 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。
• S008 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1，不以小數近似替代。
• S007 · 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。
• S006 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=2 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=2 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=2 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=2 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

(2, 2, 6, 23166, 是, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

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。