第 22512 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":4,"coefficients":[2,4,6,8],"constant":-1,"initial_values":[11,12,13,14],"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-forward-generation；資料={"order":4,"coefficients":[2,4,6,8],"constant":-1,"initial_values":[11,12,13,14],"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-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=4;oldest-to-newest coefficients=[2, 4, 6, 8];constant=-1;ALL initial a_0..a_(r-1)=[11, 12, 13, 14]

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, 11)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 22

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(-1, 22)

recurrence ORIGINAL n=4 partial sum after coefficient0 = 21

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(4, 12)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 48

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(21, 48)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 69

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(6, 13)

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 78

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(69, 78)

recurrence ORIGINAL n=4 partial sum after coefficient2 = 147

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(8, 14)

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 112

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(147, 112)

recurrence ORIGINAL n=4 partial sum after coefficient3 = 259

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[11, 12, 13, 14];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 11, 22], [1, 1, 4, 12, 48], [2, 2, 6, 13, 78], [3, 3, 8, 14, 112]]

new a_4=259;next state=[12, 13, 14, 259];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 13, 14, 259]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(2, 12)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 24

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(-1, 24)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 23

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(4, 13)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 52

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(23, 52)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 75

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(6, 14)

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 84

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(75, 84)

recurrence ORIGINAL n=5 partial sum after coefficient2 = 159

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(8, 259)

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 2072

recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(159, 2072)

recurrence ORIGINAL n=5 partial sum after coefficient3 = 2231

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[12, 13, 14, 259];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 12, 24], [1, 2, 4, 13, 52], [2, 3, 6, 14, 84], [3, 4, 8, 259, 2072]]

new a_5=2231;next state=[13, 14, 259, 2231];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 13, 14, 259, 2231]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(2, 13)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 26

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(-1, 26)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 25

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(4, 14)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 56

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(25, 56)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 81

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(6, 259)

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 1554

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(81, 1554)

recurrence ORIGINAL n=6 partial sum after coefficient2 = 1635

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(8, 2231)

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 17848

recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(1635, 17848)

recurrence ORIGINAL n=6 partial sum after coefficient3 = 19483

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[13, 14, 259, 2231];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 13, 26], [1, 3, 4, 14, 56], [2, 4, 6, 259, 1554], [3, 5, 8, 2231, 17848]]

new a_6=19483;next state=[14, 259, 2231, 19483];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 13, 14, 259, 2231, 19483]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(2, 14)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 28

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(-1, 28)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 27

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(4, 259)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 1036

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(27, 1036)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 1063

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(6, 2231)

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 13386

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(1063, 13386)

recurrence ORIGINAL n=7 partial sum after coefficient2 = 14449

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(8, 19483)

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 155864

recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(14449, 155864)

recurrence ORIGINAL n=7 partial sum after coefficient3 = 170313

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[14, 259, 2231, 19483];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 14, 28], [1, 4, 4, 259, 1036], [2, 5, 6, 2231, 13386], [3, 6, 8, 19483, 155864]]

new a_7=170313;next state=[259, 2231, 19483, 170313];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 13, 14, 259, 2231, 19483, 170313]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 := mul(2, 259)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 = 518

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(-1, 518)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 517

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 := mul(4, 2231)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 = 8924

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(517, 8924)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 9441

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 := mul(6, 19483)

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 = 116898

recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(9441, 116898)

recurrence ORIGINAL n=8 partial sum after coefficient2 = 126339

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 := mul(8, 170313)

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 = 1362504

recurrence ORIGINAL n=8 partial sum after coefficient3 := sum(126339, 1362504)

recurrence ORIGINAL n=8 partial sum after coefficient3 = 1488843

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[259, 2231, 19483, 170313];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 259, 518], [1, 5, 4, 2231, 8924], [2, 6, 6, 19483, 116898], [3, 7, 8, 170313, 1362504]]

new a_8=1488843;next state=[2231, 19483, 170313, 1488843];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 13, 14, 259, 2231, 19483, 170313, 1488843]

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 := mul(2, 2231)

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 = 4462

recurrence ORIGINAL n=9 partial sum after coefficient0 := sum(-1, 4462)

recurrence ORIGINAL n=9 partial sum after coefficient0 = 4461

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 := mul(4, 19483)

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 = 77932

recurrence ORIGINAL n=9 partial sum after coefficient1 := sum(4461, 77932)

recurrence ORIGINAL n=9 partial sum after coefficient1 = 82393

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 := mul(6, 170313)

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 = 1021878

recurrence ORIGINAL n=9 partial sum after coefficient2 := sum(82393, 1021878)

recurrence ORIGINAL n=9 partial sum after coefficient2 = 1104271

recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 := mul(8, 1488843)

recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 = 11910744

recurrence ORIGINAL n=9 partial sum after coefficient3 := sum(1104271, 11910744)

recurrence ORIGINAL n=9 partial sum after coefficient3 = 13015015

recurrence ORIGINAL:ORIGINAL target n=9;previous state a_(n-r)..a_(n-1)=[2231, 19483, 170313, 1488843];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 2231, 4462], [1, 6, 4, 19483, 77932], [2, 7, 6, 170313, 1021878], [3, 8, 8, 1488843, 11910744]]

new a_9=13015015;next state=[19483, 170313, 1488843, 13015015];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 13, 14, 259, 2231, 19483, 170313, 1488843, 13015015]

recurrence ORIGINAL n=10 ORIGINAL coefficient0 times a_6 := mul(2, 19483)

recurrence ORIGINAL n=10 ORIGINAL coefficient0 times a_6 = 38966

recurrence ORIGINAL n=10 partial sum after coefficient0 := sum(-1, 38966)

recurrence ORIGINAL n=10 partial sum after coefficient0 = 38965

recurrence ORIGINAL n=10 ORIGINAL coefficient1 times a_7 := mul(4, 170313)

recurrence ORIGINAL n=10 ORIGINAL coefficient1 times a_7 = 681252

recurrence ORIGINAL n=10 partial sum after coefficient1 := sum(38965, 681252)

recurrence ORIGINAL n=10 partial sum after coefficient1 = 720217

recurrence ORIGINAL n=10 ORIGINAL coefficient2 times a_8 := mul(6, 1488843)

recurrence ORIGINAL n=10 ORIGINAL coefficient2 times a_8 = 8933058

recurrence ORIGINAL n=10 partial sum after coefficient2 := sum(720217, 8933058)

recurrence ORIGINAL n=10 partial sum after coefficient2 = 9653275

recurrence ORIGINAL n=10 ORIGINAL coefficient3 times a_9 := mul(8, 13015015)

recurrence ORIGINAL n=10 ORIGINAL coefficient3 times a_9 = 104120120

recurrence ORIGINAL n=10 partial sum after coefficient3 := sum(9653275, 104120120)

recurrence ORIGINAL n=10 partial sum after coefficient3 = 113773395

recurrence ORIGINAL:ORIGINAL target n=10;previous state a_(n-r)..a_(n-1)=[19483, 170313, 1488843, 13015015];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 2, 19483, 38966], [1, 7, 4, 170313, 681252], [2, 8, 6, 1488843, 8933058], [3, 9, 8, 13015015, 104120120]]

new a_10=113773395;next state=[170313, 1488843, 13015015, 113773395];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 13, 14, 259, 2231, 19483, 170313, 1488843, 13015015, 113773395]

recurrence ORIGINAL:complete ORIGINAL initial state=[11, 12, 13, 14];actual generated prefix=[11, 12, 13, 14, 259, 2231, 19483, 170313, 1488843, 13015015, 113773395]

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, [11, 12, 13, 14], 10, [11, 12, 13, 14, 259, 2231, 19483, 170313, 1488843, 13015015, 113773395], 113773395, 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。