第 22513 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":2,"coefficients":[1,2],"constant":0,"initial_values":[12,13],"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-forward-generation；資料={"order":2,"coefficients":[1,2],"constant":0,"initial_values":[12,13],"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-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=2;oldest-to-newest coefficients=[1, 2];constant=0;ALL initial a_0..a_(r-1)=[12, 13]

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, 12)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 12

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(0, 12)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 12

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(2, 13)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 26

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(12, 26)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 38

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[12, 13];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 12, 12], [1, 1, 2, 13, 26]]

new a_2=38;next state=[13, 38];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(1, 13)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 13

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(0, 13)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 13

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(2, 38)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 76

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(13, 76)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 89

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[13, 38];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 13, 13], [1, 2, 2, 38, 76]]

new a_3=89;next state=[38, 89];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38, 89]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(1, 38)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 38

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(0, 38)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 38

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(2, 89)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 178

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(38, 178)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 216

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[38, 89];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 38, 38], [1, 3, 2, 89, 178]]

new a_4=216;next state=[89, 216];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38, 89, 216]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(1, 89)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 89

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(0, 89)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 89

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(2, 216)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 432

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(89, 432)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 521

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[89, 216];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 89, 89], [1, 4, 2, 216, 432]]

new a_5=521;next state=[216, 521];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38, 89, 216, 521]

recurrence ORIGINAL:complete ORIGINAL initial state=[12, 13];actual generated prefix=[12, 13, 38, 89, 216, 521]

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, [12, 13], 5, [12, 13, 38, 89, 216, 521], 521, 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。