第 22544 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":1,"initial_values":[18,19],"target_n":7}。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":1,"initial_values":[18,19],"target_n":7}。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=1;ALL initial a_0..a_(r-1)=[18, 19]

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, 18)

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 18

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(1, 18)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 19

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(2, 19)

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 38

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(19, 38)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 57

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[18, 19];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 18, 18], [1, 1, 2, 19, 38]]

new a_2=57;next state=[19, 57];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(1, 19)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 19

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(1, 19)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 20

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(2, 57)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 114

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(20, 114)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 134

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[19, 57];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 19, 19], [1, 2, 2, 57, 114]]

new a_3=134;next state=[57, 134];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57, 134]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(1, 57)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 57

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(1, 57)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 58

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(2, 134)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 268

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(58, 268)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 326

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[57, 134];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 57, 57], [1, 3, 2, 134, 268]]

new a_4=326;next state=[134, 326];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57, 134, 326]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(1, 134)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 134

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(1, 134)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 135

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(2, 326)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 652

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(135, 652)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 787

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[134, 326];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 134, 134], [1, 4, 2, 326, 652]]

new a_5=787;next state=[326, 787];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57, 134, 326, 787]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 := mul(1, 326)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 = 326

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(1, 326)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 327

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 := mul(2, 787)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 = 1574

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(327, 1574)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 1901

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[326, 787];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 326, 326], [1, 5, 2, 787, 1574]]

new a_6=1901;next state=[787, 1901];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57, 134, 326, 787, 1901]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 := mul(1, 787)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 = 787

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(1, 787)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 788

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 := mul(2, 1901)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 = 3802

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(788, 3802)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 4590

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[787, 1901];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 1, 787, 787], [1, 6, 2, 1901, 3802]]

new a_7=4590;next state=[1901, 4590];drop ONLY oldest state entry then append new scalar;prefix now=[18, 19, 57, 134, 326, 787, 1901, 4590]

recurrence ORIGINAL:complete ORIGINAL initial state=[18, 19];actual generated prefix=[18, 19, 57, 134, 326, 787, 1901, 4590]

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

• S033 · 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。
• S032 · 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。
• S031 · 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。
• S030 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6，不以小數近似替代。
• S029 · 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。
• S028 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5，不以小數近似替代。
• 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, 7, 4590, 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。