第 22543 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":[2,4,6,8],"constant":0,"initial_values":[17,18,19,20],"target_n":8}。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":[2,4,6,8],"constant":0,"initial_values":[17,18,19,20],"target_n":8}。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=[2, 4, 6, 8];constant=0;ALL initial a_0..a_(r-1)=[17, 18, 19, 20]

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(2, 17)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 34

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(0, 34)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 34

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(4, 18)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 72

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(34, 72)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 106

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(6, 19)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 114

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(106, 114)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 220

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(8, 20)

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 160

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(220, 160)

recurrence ORIGINAL n=4 partial sum after coefficient3 = 380

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[17, 18, 19, 20];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 17, 34], [1, 1, 4, 18, 72], [2, 2, 6, 19, 114], [3, 3, 8, 20, 160]]

new a_4=380;next state=[18, 19, 20, 380];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 19, 20, 380]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(2, 18)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 36

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(0, 36)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 36

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(4, 19)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 76

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(36, 76)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 112

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(6, 20)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 120

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(112, 120)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 232

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(8, 380)

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 3040

recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(232, 3040)

recurrence ORIGINAL n=5 partial sum after coefficient3 = 3272

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[18, 19, 20, 380];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 18, 36], [1, 2, 4, 19, 76], [2, 3, 6, 20, 120], [3, 4, 8, 380, 3040]]

new a_5=3272;next state=[19, 20, 380, 3272];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 19, 20, 380, 3272]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(2, 19)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 38

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(0, 38)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 38

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(4, 20)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 80

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(38, 80)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 118

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(6, 380)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 2280

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(118, 2280)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 2398

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(8, 3272)

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 26176

recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(2398, 26176)

recurrence ORIGINAL n=6 partial sum after coefficient3 = 28574

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[19, 20, 380, 3272];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 19, 38], [1, 3, 4, 20, 80], [2, 4, 6, 380, 2280], [3, 5, 8, 3272, 26176]]

new a_6=28574;next state=[20, 380, 3272, 28574];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 19, 20, 380, 3272, 28574]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(2, 20)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 40

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(0, 40)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 40

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(4, 380)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 1520

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(40, 1520)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 1560

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(6, 3272)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 19632

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(1560, 19632)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 21192

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(8, 28574)

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 228592

recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(21192, 228592)

recurrence ORIGINAL n=7 partial sum after coefficient3 = 249784

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[20, 380, 3272, 28574];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 20, 40], [1, 4, 4, 380, 1520], [2, 5, 6, 3272, 19632], [3, 6, 8, 28574, 228592]]

new a_7=249784;next state=[380, 3272, 28574, 249784];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 19, 20, 380, 3272, 28574, 249784]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 := mul(2, 380)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 = 760

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(0, 760)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 760

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 := mul(4, 3272)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 = 13088

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(760, 13088)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 13848

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 := mul(6, 28574)

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 = 171444

recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(13848, 171444)

recurrence ORIGINAL n=8 partial sum after coefficient2 = 185292

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 := mul(8, 249784)

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 = 1998272

recurrence ORIGINAL n=8 partial sum after coefficient3 := sum(185292, 1998272)

recurrence ORIGINAL n=8 partial sum after coefficient3 = 2183564

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[380, 3272, 28574, 249784];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 380, 760], [1, 5, 4, 3272, 13088], [2, 6, 6, 28574, 171444], [3, 7, 8, 249784, 1998272]]

new a_8=2183564;next state=[3272, 28574, 249784, 2183564];drop ONLY oldest state entry then append new scalar;prefix now=[17, 18, 19, 20, 380, 3272, 28574, 249784, 2183564]

recurrence ORIGINAL:complete ORIGINAL initial state=[17, 18, 19, 20];actual generated prefix=[17, 18, 19, 20, 380, 3272, 28574, 249784, 2183564]

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

• S048 · 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。
• S047 · 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。
• S046 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 partial sum after coefficient3，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 partial sum after coefficient2，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 partial sum after coefficient1，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 partial sum after coefficient0，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4，不以小數近似替代。
• 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, 8, 2183564, 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。