第 22534 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":4,"coefficients":[1,2,3,4],"constant":1,"initial_values":[8,9,10,11],"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":4,"coefficients":[1,2,3,4],"constant":1,"initial_values":[8,9,10,11],"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=4;oldest-to-newest coefficients=[1, 2, 3, 4];constant=1;ALL initial a_0..a_(r-1)=[8, 9, 10, 11]

for EVERY n>=4,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=4;forward rule is total and single-valued

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 := mul(1, 8)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 8

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(1, 8)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 9

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(2, 9)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 18

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(9, 18)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 27

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(3, 10)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 30

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(27, 30)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 57

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(4, 11)

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 44

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(57, 44)

recurrence ORIGINAL n=4 partial sum after coefficient3 = 101

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[8, 9, 10, 11];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 8, 8], [1, 1, 2, 9, 18], [2, 2, 3, 10, 30], [3, 3, 4, 11, 44]]

new a_4=101;next state=[9, 10, 11, 101];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 10, 11, 101]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(1, 9)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 9

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(1, 9)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 10

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(2, 10)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 20

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(10, 20)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 30

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(3, 11)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 33

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(30, 33)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 63

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(4, 101)

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 404

recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(63, 404)

recurrence ORIGINAL n=5 partial sum after coefficient3 = 467

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[9, 10, 11, 101];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 9, 9], [1, 2, 2, 10, 20], [2, 3, 3, 11, 33], [3, 4, 4, 101, 404]]

new a_5=467;next state=[10, 11, 101, 467];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 10, 11, 101, 467]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(1, 10)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 10

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(1, 10)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 11

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(2, 11)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 22

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(11, 22)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 33

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(3, 101)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 303

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(33, 303)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 336

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(4, 467)

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 1868

recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(336, 1868)

recurrence ORIGINAL n=6 partial sum after coefficient3 = 2204

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[10, 11, 101, 467];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 10, 10], [1, 3, 2, 11, 22], [2, 4, 3, 101, 303], [3, 5, 4, 467, 1868]]

new a_6=2204;next state=[11, 101, 467, 2204];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 10, 11, 101, 467, 2204]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(1, 11)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 11

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(1, 11)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 12

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(2, 101)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 202

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(12, 202)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 214

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(3, 467)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 1401

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(214, 1401)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 1615

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(4, 2204)

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 8816

recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(1615, 8816)

recurrence ORIGINAL n=7 partial sum after coefficient3 = 10431

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[11, 101, 467, 2204];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 11, 11], [1, 4, 2, 101, 202], [2, 5, 3, 467, 1401], [3, 6, 4, 2204, 8816]]

new a_7=10431;next state=[101, 467, 2204, 10431];drop ONLY oldest state entry then append new scalar;prefix now=[8, 9, 10, 11, 101, 467, 2204, 10431]

recurrence ORIGINAL:complete ORIGINAL initial state=[8, 9, 10, 11];actual generated prefix=[8, 9, 10, 11, 101, 467, 2204, 10431]

ANY two sequences with SAME r initials and SAME total rule agree at n=4 by substitution,then at every later n by induction;this determines the requested prefix only

逐步反向核對

• S039 · 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。
• S038 · 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。
• S037 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 partial sum after coefficient3，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6，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 partial sum after coefficient2，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 ORIGINAL coefficient2 times a_5，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 partial sum after coefficient1，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 ORIGINAL coefficient1 times a_4，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 partial sum after coefficient0，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 coefficient0 times a_3，不以小數近似替代。
• S029 · 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。
• S028 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 partial sum after coefficient3，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 ORIGINAL coefficient3 times a_5，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 partial sum after coefficient2，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 coefficient2 times a_4，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 coefficient1，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 coefficient1 times a_3，不以小數近似替代。
• S022 · 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。
• S021 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2，不以小數近似替代。
• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 partial sum after coefficient3，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 coefficient3 times a_4，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 partial sum after coefficient2，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3，不以小數近似替代。
• S015 · 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。
• S014 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2，不以小數近似替代。
• S013 · 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。
• S012 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1，不以小數近似替代。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 partial sum after coefficient3，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3，不以小數近似替代。
• S008 · 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。
• S007 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 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=4 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=4 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=4 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=4 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

(4, 4, 7, 10431, 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。