第 22550 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":[1,2],"constant":2,"initial_values":[24,25],"target_n":5}。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":[1,2],"constant":2,"initial_values":[24,25],"target_n":5}。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=[1, 2];constant=2;ALL initial a_0..a_(r-1)=[24, 25]

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(1, 24)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 24

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(2, 24)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 26

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(2, 25)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 50

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(26, 50)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 76

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[24, 25];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 24, 24], [1, 1, 2, 25, 50]]

new a_2=76;next state=[25, 76];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 76]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(1, 25)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 25

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(2, 25)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 27

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(2, 76)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 152

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(27, 152)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 179

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[25, 76];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 25, 25], [1, 2, 2, 76, 152]]

new a_3=179;next state=[76, 179];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 76, 179]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(1, 76)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 76

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(2, 76)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 78

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(2, 179)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 358

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(78, 358)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 436

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[76, 179];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 76, 76], [1, 3, 2, 179, 358]]

new a_4=436;next state=[179, 436];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 76, 179, 436]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(1, 179)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 179

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(2, 179)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 181

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(2, 436)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 872

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(181, 872)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 1053

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[179, 436];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 179, 179], [1, 4, 2, 436, 872]]

new a_5=1053;next state=[436, 1053];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 76, 179, 436, 1053]

recurrence ORIGINAL:complete ORIGINAL initial state=[24, 25];actual generated prefix=[24, 25, 76, 179, 436, 1053]

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

• 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 — 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, 5, 1053, 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。