第 22514 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":3,"coefficients":[2,4,6],"constant":1,"initial_values":[13,14,15],"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-forward-generation；資料={"order":3,"coefficients":[2,4,6],"constant":1,"initial_values":[13,14,15],"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-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=3;oldest-to-newest coefficients=[2, 4, 6];constant=1;ALL initial a_0..a_(r-1)=[13, 14, 15]

for EVERY n>=3,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=3;forward rule is total and single-valued

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 := mul(2, 13)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 = 26

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(1, 26)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 27

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 := mul(4, 14)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 = 56

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(27, 56)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 83

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 := mul(6, 15)

recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 = 90

recurrence ORIGINAL n=3 partial sum after coefficient2 := sum(83, 90)

recurrence ORIGINAL n=3 partial sum after coefficient2 = 173

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[13, 14, 15];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 13, 26], [1, 1, 4, 14, 56], [2, 2, 6, 15, 90]]

new a_3=173;next state=[14, 15, 173];drop ONLY oldest state entry then append new scalar;prefix now=[13, 14, 15, 173]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 := mul(2, 14)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1 = 28

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(1, 28)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 29

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 := mul(4, 15)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2 = 60

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(29, 60)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 89

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 := mul(6, 173)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3 = 1038

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(89, 1038)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 1127

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[14, 15, 173];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 14, 28], [1, 2, 4, 15, 60], [2, 3, 6, 173, 1038]]

new a_4=1127;next state=[15, 173, 1127];drop ONLY oldest state entry then append new scalar;prefix now=[13, 14, 15, 173, 1127]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 := mul(2, 15)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_2 = 30

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(1, 30)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 31

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 := mul(4, 173)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_3 = 692

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(31, 692)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 723

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 := mul(6, 1127)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4 = 6762

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(723, 6762)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 7485

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[15, 173, 1127];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 15, 30], [1, 3, 4, 173, 692], [2, 4, 6, 1127, 6762]]

new a_5=7485;next state=[173, 1127, 7485];drop ONLY oldest state entry then append new scalar;prefix now=[13, 14, 15, 173, 1127, 7485]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 := mul(2, 173)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_3 = 346

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(1, 346)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 347

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 := mul(4, 1127)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_4 = 4508

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(347, 4508)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 4855

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 := mul(6, 7485)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5 = 44910

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(4855, 44910)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 49765

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[173, 1127, 7485];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 173, 346], [1, 4, 4, 1127, 4508], [2, 5, 6, 7485, 44910]]

new a_6=49765;next state=[1127, 7485, 49765];drop ONLY oldest state entry then append new scalar;prefix now=[13, 14, 15, 173, 1127, 7485, 49765]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 := mul(2, 1127)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_4 = 2254

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(1, 2254)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 2255

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 := mul(4, 7485)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_5 = 29940

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(2255, 29940)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 32195

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 := mul(6, 49765)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6 = 298590

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(32195, 298590)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 330785

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[1127, 7485, 49765];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 1127, 2254], [1, 5, 4, 7485, 29940], [2, 6, 6, 49765, 298590]]

new a_7=330785;next state=[7485, 49765, 330785];drop ONLY oldest state entry then append new scalar;prefix now=[13, 14, 15, 173, 1127, 7485, 49765, 330785]

recurrence ORIGINAL:complete ORIGINAL initial state=[13, 14, 15];actual generated prefix=[13, 14, 15, 173, 1127, 7485, 49765, 330785]

ANY two sequences with SAME r initials and SAME total rule agree at n=3 by substitution,then at every later n by induction;this determines the requested prefix only

逐步反向核對

• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S036 · 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。
• S035 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_6，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 partial sum after coefficient1，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 ORIGINAL coefficient1 times a_5，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 partial sum after coefficient0，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 ORIGINAL coefficient0 times a_4，不以小數近似替代。
• S030 · 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。
• S029 · 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。
• S028 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_5，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 partial sum after coefficient1，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 ORIGINAL coefficient1 times a_4，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 partial sum after coefficient0，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 ORIGINAL coefficient0 times a_3，不以小數近似替代。
• 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 — 以精確有理數重算 recurrence ORIGINAL n=5 partial sum after coefficient2，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_4，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 partial sum after coefficient1，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 ORIGINAL coefficient1 times a_3，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 partial sum after coefficient0，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 ORIGINAL coefficient0 times a_2，不以小數近似替代。
• S016 · 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。
• S015 · 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。
• S014 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_3，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 partial sum after coefficient1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_2，不以小數近似替代。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_1，不以小數近似替代。
• S009 · 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。
• S008 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=3 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=3 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=3 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=3 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=3 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=3 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

(3, [13, 14, 15], 7, [13, 14, 15, 173, 1127, 7485, 49765, 330785], 330785, 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。