第 22575 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-missing-initial-boundary；資料={"coefficients":[1,6],"constant":0,"first_initial":24,"second_left":25,"second_right":26,"target_n":6}。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-missing-initial-boundary；資料={"coefficients":[1,6],"constant":0,"first_initial":24,"second_left":25,"second_right":26,"target_n":6}。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-missing-initial-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary]

recurrence LEFT completion:ORIGINAL order=2;oldest-to-newest coefficients=[1, 6];constant=0;ALL initial a_0..a_(r-1)=[24, 25]

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 LEFT completion n=2 ORIGINAL coefficient0 times a_0 := mul(1, 24)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 24

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(0, 24)

recurrence LEFT completion n=2 partial sum after coefficient0 = 24

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(6, 25)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 150

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(24, 150)

recurrence LEFT completion n=2 partial sum after coefficient1 = 174

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[24, 25];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 24, 24], [1, 1, 6, 25, 150]]

new a_2=174;next state=[25, 174];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 174]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 25)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 25

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(0, 25)

recurrence LEFT completion n=3 partial sum after coefficient0 = 25

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(6, 174)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 1044

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(25, 1044)

recurrence LEFT completion n=3 partial sum after coefficient1 = 1069

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[25, 174];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 25, 25], [1, 2, 6, 174, 1044]]

new a_3=1069;next state=[174, 1069];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 174, 1069]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 174)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 174

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(0, 174)

recurrence LEFT completion n=4 partial sum after coefficient0 = 174

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(6, 1069)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 6414

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(174, 6414)

recurrence LEFT completion n=4 partial sum after coefficient1 = 6588

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[174, 1069];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 174, 174], [1, 3, 6, 1069, 6414]]

new a_4=6588;next state=[1069, 6588];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 174, 1069, 6588]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 1069)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 1069

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(0, 1069)

recurrence LEFT completion n=5 partial sum after coefficient0 = 1069

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(6, 6588)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 39528

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(1069, 39528)

recurrence LEFT completion n=5 partial sum after coefficient1 = 40597

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[1069, 6588];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 1069, 1069], [1, 4, 6, 6588, 39528]]

new a_5=40597;next state=[6588, 40597];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 174, 1069, 6588, 40597]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 6588)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 6588

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(0, 6588)

recurrence LEFT completion n=6 partial sum after coefficient0 = 6588

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(6, 40597)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 243582

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(6588, 243582)

recurrence LEFT completion n=6 partial sum after coefficient1 = 250170

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[6588, 40597];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 6588, 6588], [1, 5, 6, 40597, 243582]]

new a_6=250170;next state=[40597, 250170];drop ONLY oldest state entry then append new scalar;prefix now=[24, 25, 174, 1069, 6588, 40597, 250170]

recurrence LEFT completion:complete ORIGINAL initial state=[24, 25];actual generated prefix=[24, 25, 174, 1069, 6588, 40597, 250170]

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

recurrence RIGHT completion:ORIGINAL order=2;oldest-to-newest coefficients=[1, 6];constant=0;ALL initial a_0..a_(r-1)=[24, 26]

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 RIGHT completion n=2 ORIGINAL coefficient0 times a_0 := mul(1, 24)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 24

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(0, 24)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 24

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(6, 26)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 156

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(24, 156)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 180

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[24, 26];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 24, 24], [1, 1, 6, 26, 156]]

new a_2=180;next state=[26, 180];drop ONLY oldest state entry then append new scalar;prefix now=[24, 26, 180]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 26)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 26

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(0, 26)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 26

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(6, 180)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 1080

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(26, 1080)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 1106

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[26, 180];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 26, 26], [1, 2, 6, 180, 1080]]

new a_3=1106;next state=[180, 1106];drop ONLY oldest state entry then append new scalar;prefix now=[24, 26, 180, 1106]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 180)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 180

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(0, 180)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 180

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(6, 1106)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 6636

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(180, 6636)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 6816

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[180, 1106];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 180, 180], [1, 3, 6, 1106, 6636]]

new a_4=6816;next state=[1106, 6816];drop ONLY oldest state entry then append new scalar;prefix now=[24, 26, 180, 1106, 6816]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 1106)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 1106

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(0, 1106)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 1106

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(6, 6816)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 40896

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(1106, 40896)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 42002

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[1106, 6816];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 1106, 1106], [1, 4, 6, 6816, 40896]]

new a_5=42002;next state=[6816, 42002];drop ONLY oldest state entry then append new scalar;prefix now=[24, 26, 180, 1106, 6816, 42002]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 6816)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 6816

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(0, 6816)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 6816

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(6, 42002)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 252012

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(6816, 252012)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 258828

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[6816, 42002];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 6816, 6816], [1, 5, 6, 42002, 252012]]

new a_6=258828;next state=[42002, 258828];drop ONLY oldest state entry then append new scalar;prefix now=[24, 26, 180, 1106, 6816, 42002, 258828]

recurrence RIGHT completion:complete ORIGINAL initial state=[24, 26];actual generated prefix=[24, 26, 180, 1106, 6816, 42002, 258828]

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

ORIGINAL only known a_0=24;different completions a_1=25 and26;target=6

WHOLE prefixes including initials differ=是;actual targets=[250170, 258828] MAY coincide;one_initial_uniqueness_claim fixed No

逐步反向核對

• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S054 · 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。
• S053 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=6 partial sum after coefficient1，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=6 partial sum after coefficient0，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4，不以小數近似替代。
• S049 · 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。
• S048 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=5 partial sum after coefficient1，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=5 partial sum after coefficient0，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3，不以小數近似替代。
• S044 · 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。
• S043 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=4 partial sum after coefficient1，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=4 partial sum after coefficient0，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2，不以小數近似替代。
• S039 · 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。
• S038 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=3 partial sum after coefficient1，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=3 partial sum after coefficient0，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1，不以小數近似替代。
• S034 · 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。
• S033 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=2 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 RIGHT completion n=2 ORIGINAL coefficient1 times a_1，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=2 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 RIGHT completion n=2 ORIGINAL coefficient0 times a_0，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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 LEFT completion 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

(24, 25, 26, [24, 25, 174, 1069, 6588, 40597, 250170], [24, 26, 180, 1106, 6816, 42002, 258828], 是, 否, order_two_recurrence_with_only_one_initial_value_is_not_unique)

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。