第 22529 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":2,"coefficients":[2,4],"constant":1,"initial_values":[3,4],"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":2,"coefficients":[2,4],"constant":1,"initial_values":[3,4],"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=2;oldest-to-newest coefficients=[2, 4];constant=1;ALL initial a_0..a_(r-1)=[3, 4]

for EVERY n>=2,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=2;forward rule is total and single-valued

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

recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 6

recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(1, 6)

recurrence ORIGINAL n=2 partial sum after coefficient0 = 7

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

recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 16

recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(7, 16)

recurrence ORIGINAL n=2 partial sum after coefficient1 = 23

recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[3, 4];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 2, 3, 6], [1, 1, 4, 4, 16]]

new a_2=23;next state=[4, 23];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23]

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(2, 4)

recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 8

recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(1, 8)

recurrence ORIGINAL n=3 partial sum after coefficient0 = 9

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(4, 23)

recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 92

recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(9, 92)

recurrence ORIGINAL n=3 partial sum after coefficient1 = 101

recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[4, 23];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 2, 4, 8], [1, 2, 4, 23, 92]]

new a_3=101;next state=[23, 101];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101]

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(2, 23)

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 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_3 := mul(4, 101)

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 404

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(47, 404)

recurrence ORIGINAL n=4 partial sum after coefficient1 = 451

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[23, 101];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 2, 23, 46], [1, 3, 4, 101, 404]]

new a_4=451;next state=[101, 451];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101, 451]

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(2, 101)

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 202

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(1, 202)

recurrence ORIGINAL n=5 partial sum after coefficient0 = 203

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(4, 451)

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 1804

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(203, 1804)

recurrence ORIGINAL n=5 partial sum after coefficient1 = 2007

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[101, 451];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 2, 101, 202], [1, 4, 4, 451, 1804]]

new a_5=2007;next state=[451, 2007];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101, 451, 2007]

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 := mul(2, 451)

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_4 = 902

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(1, 902)

recurrence ORIGINAL n=6 partial sum after coefficient0 = 903

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 := mul(4, 2007)

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5 = 8028

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(903, 8028)

recurrence ORIGINAL n=6 partial sum after coefficient1 = 8931

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[451, 2007];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 2, 451, 902], [1, 5, 4, 2007, 8028]]

new a_6=8931;next state=[2007, 8931];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101, 451, 2007, 8931]

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 := mul(2, 2007)

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5 = 4014

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(1, 4014)

recurrence ORIGINAL n=7 partial sum after coefficient0 = 4015

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 := mul(4, 8931)

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6 = 35724

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(4015, 35724)

recurrence ORIGINAL n=7 partial sum after coefficient1 = 39739

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[2007, 8931];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 2, 2007, 4014], [1, 6, 4, 8931, 35724]]

new a_7=39739;next state=[8931, 39739];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101, 451, 2007, 8931, 39739]

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_6 := mul(2, 8931)

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_6 = 17862

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(1, 17862)

recurrence ORIGINAL n=8 partial sum after coefficient0 = 17863

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_7 := mul(4, 39739)

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_7 = 158956

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(17863, 158956)

recurrence ORIGINAL n=8 partial sum after coefficient1 = 176819

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[8931, 39739];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 2, 8931, 17862], [1, 7, 4, 39739, 158956]]

new a_8=176819;next state=[39739, 176819];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 23, 101, 451, 2007, 8931, 39739, 176819]

recurrence ORIGINAL:complete ORIGINAL initial state=[3, 4];actual generated prefix=[3, 4, 23, 101, 451, 2007, 8931, 39739, 176819]

ANY two sequences with SAME r initials and SAME total rule agree at n=2 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=8 partial sum after coefficient1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_7，不以小數近似替代。
• S034 · 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。
• S033 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_6，不以小數近似替代。
• S032 · 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。
• S031 · 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。
• S030 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_6，不以小數近似替代。
• S029 · 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。
• S028 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_5，不以小數近似替代。
• S027 · 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。
• S026 · 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。
• S025 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_5，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 coefficient0，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 coefficient0 times a_4，不以小數近似替代。
• S022 · 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。
• S021 · 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。
• S020 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4，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 partial sum after coefficient0，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 coefficient0 times a_3，不以小數近似替代。
• S017 · 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。
• S016 · 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。
• S015 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3，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 partial sum after coefficient0，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 ORIGINAL coefficient0 times a_2，不以小數近似替代。
• S012 · 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。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2，不以小數近似替代。
• S009 · 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。
• S008 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1，不以小數近似替代。
• S007 · 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。
• S006 · premise_and_result_audit — 以精確有理數重算 recurrence ORIGINAL n=2 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=2 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=2 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=2 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

(2, 2, 8, 176819, 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。