第 22549 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":1,"initial_values":[23,24,25,26],"target_n":10}。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":1,"initial_values":[23,24,25,26],"target_n":10}。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=1;ALL initial a_0..a_(r-1)=[23, 24, 25, 26]

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, 23)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 46

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(1, 46)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 47

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(4, 24)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 96

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(47, 96)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 143

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(6, 25)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 150

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(143, 150)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 293

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(8, 26)

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 208

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(293, 208)

recurrence ORIGINAL n=4 partial sum after coefficient3 = 501

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[23, 24, 25, 26];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 23, 46], [1, 1, 4, 24, 96], [2, 2, 6, 25, 150], [3, 3, 8, 26, 208]]

new a_4=501;next state=[24, 25, 26, 501];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 25, 26, 501]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(2, 24)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 48

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(1, 48)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 49

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(4, 25)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 100

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(49, 100)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 149

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(6, 26)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 156

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(149, 156)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 305

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(8, 501)

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 4008

recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(305, 4008)

recurrence ORIGINAL n=5 partial sum after coefficient3 = 4313

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[24, 25, 26, 501];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 24, 48], [1, 2, 4, 25, 100], [2, 3, 6, 26, 156], [3, 4, 8, 501, 4008]]

new a_5=4313;next state=[25, 26, 501, 4313];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 25, 26, 501, 4313]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(2, 25)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 50

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(1, 50)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 51

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(4, 26)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 104

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(51, 104)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 155

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(6, 501)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 3006

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(155, 3006)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 3161

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(8, 4313)

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 34504

recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(3161, 34504)

recurrence ORIGINAL n=6 partial sum after coefficient3 = 37665

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[25, 26, 501, 4313];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 25, 50], [1, 3, 4, 26, 104], [2, 4, 6, 501, 3006], [3, 5, 8, 4313, 34504]]

new a_6=37665;next state=[26, 501, 4313, 37665];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 25, 26, 501, 4313, 37665]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(2, 26)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 52

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(1, 52)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 53

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(4, 501)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 2004

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(53, 2004)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 2057

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(6, 4313)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 25878

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(2057, 25878)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 27935

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(8, 37665)

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 301320

recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(27935, 301320)

recurrence ORIGINAL n=7 partial sum after coefficient3 = 329255

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[26, 501, 4313, 37665];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 26, 52], [1, 4, 4, 501, 2004], [2, 5, 6, 4313, 25878], [3, 6, 8, 37665, 301320]]

new a_7=329255;next state=[501, 4313, 37665, 329255];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 25, 26, 501, 4313, 37665, 329255]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 := mul(2, 501)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 = 1002

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(1, 1002)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 1003

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 := mul(4, 4313)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 = 17252

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(1003, 17252)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 18255

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 := mul(6, 37665)

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 = 225990

recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(18255, 225990)

recurrence ORIGINAL n=8 partial sum after coefficient2 = 244245

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 := mul(8, 329255)

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 = 2634040

recurrence ORIGINAL n=8 partial sum after coefficient3 := sum(244245, 2634040)

recurrence ORIGINAL n=8 partial sum after coefficient3 = 2878285

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[501, 4313, 37665, 329255];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 501, 1002], [1, 5, 4, 4313, 17252], [2, 6, 6, 37665, 225990], [3, 7, 8, 329255, 2634040]]

new a_8=2878285;next state=[4313, 37665, 329255, 2878285];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 25, 26, 501, 4313, 37665, 329255, 2878285]

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 := mul(2, 4313)

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 = 8626

recurrence ORIGINAL n=9 partial sum after coefficient0 := sum(1, 8626)

recurrence ORIGINAL n=9 partial sum after coefficient0 = 8627

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 := mul(4, 37665)

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 = 150660

recurrence ORIGINAL n=9 partial sum after coefficient1 := sum(8627, 150660)

recurrence ORIGINAL n=9 partial sum after coefficient1 = 159287

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 := mul(6, 329255)

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 = 1975530

recurrence ORIGINAL n=9 partial sum after coefficient2 := sum(159287, 1975530)

recurrence ORIGINAL n=9 partial sum after coefficient2 = 2134817

recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 := mul(8, 2878285)

recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 = 23026280

recurrence ORIGINAL n=9 partial sum after coefficient3 := sum(2134817, 23026280)

recurrence ORIGINAL n=9 partial sum after coefficient3 = 25161097

recurrence ORIGINAL:ORIGINAL target n=9;previous state a_(n-r)..a_(n-1)=[4313, 37665, 329255, 2878285];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 4313, 8626], [1, 6, 4, 37665, 150660], [2, 7, 6, 329255, 1975530], [3, 8, 8, 2878285, 23026280]]

new a_9=25161097;next state=[37665, 329255, 2878285, 25161097];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 25, 26, 501, 4313, 37665, 329255, 2878285, 25161097]

recurrence ORIGINAL n=10 ORIGINAL coefficient0 times a_6 := mul(2, 37665)

recurrence ORIGINAL n=10 ORIGINAL coefficient0 times a_6 = 75330

recurrence ORIGINAL n=10 partial sum after coefficient0 := sum(1, 75330)

recurrence ORIGINAL n=10 partial sum after coefficient0 = 75331

recurrence ORIGINAL n=10 ORIGINAL coefficient1 times a_7 := mul(4, 329255)

recurrence ORIGINAL n=10 ORIGINAL coefficient1 times a_7 = 1317020

recurrence ORIGINAL n=10 partial sum after coefficient1 := sum(75331, 1317020)

recurrence ORIGINAL n=10 partial sum after coefficient1 = 1392351

recurrence ORIGINAL n=10 ORIGINAL coefficient2 times a_8 := mul(6, 2878285)

recurrence ORIGINAL n=10 ORIGINAL coefficient2 times a_8 = 17269710

recurrence ORIGINAL n=10 partial sum after coefficient2 := sum(1392351, 17269710)

recurrence ORIGINAL n=10 partial sum after coefficient2 = 18662061

recurrence ORIGINAL n=10 ORIGINAL coefficient3 times a_9 := mul(8, 25161097)

recurrence ORIGINAL n=10 ORIGINAL coefficient3 times a_9 = 201288776

recurrence ORIGINAL n=10 partial sum after coefficient3 := sum(18662061, 201288776)

recurrence ORIGINAL n=10 partial sum after coefficient3 = 219950837

recurrence ORIGINAL:ORIGINAL target n=10;previous state a_(n-r)..a_(n-1)=[37665, 329255, 2878285, 25161097];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 2, 37665, 75330], [1, 7, 4, 329255, 1317020], [2, 8, 6, 2878285, 17269710], [3, 9, 8, 25161097, 201288776]]

new a_10=219950837;next state=[329255, 2878285, 25161097, 219950837];drop ONLY oldest state entry then append new scalar;prefix now=[23, 24, 25, 26, 501, 4313, 37665, 329255, 2878285, 25161097, 219950837]

recurrence ORIGINAL:complete ORIGINAL initial state=[23, 24, 25, 26];actual generated prefix=[23, 24, 25, 26, 501, 4313, 37665, 329255, 2878285, 25161097, 219950837]

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

• S066 · 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。
• S065 · 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。
• S064 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=10 partial sum after coefficient3，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=10 ORIGINAL coefficient3 times a_9，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=10 partial sum after coefficient2，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=10 ORIGINAL coefficient2 times a_8，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=10 partial sum after coefficient1，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=10 ORIGINAL coefficient1 times a_7，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=10 partial sum after coefficient0，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=10 ORIGINAL coefficient0 times a_6，不以小數近似替代。
• S056 · 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。
• S055 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 partial sum after coefficient3，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 partial sum after coefficient2，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 partial sum after coefficient1，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 partial sum after coefficient0，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5，不以小數近似替代。
• 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, 10, 219950837, 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。