第 22563 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":[1,4],"constant":0,"first_initial":12,"second_left":13,"second_right":14,"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":[1,4],"constant":0,"first_initial":12,"second_left":13,"second_right":14,"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, 4];constant=0;ALL initial a_0..a_(r-1)=[12, 13]

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, 12)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 12

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(0, 12)

recurrence LEFT completion n=2 partial sum after coefficient0 = 12

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(4, 13)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 52

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(12, 52)

recurrence LEFT completion n=2 partial sum after coefficient1 = 64

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[12, 13];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 12, 12], [1, 1, 4, 13, 52]]

new a_2=64;next state=[13, 64];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 64]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 13)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 13

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(0, 13)

recurrence LEFT completion n=3 partial sum after coefficient0 = 13

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 64)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 256

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(13, 256)

recurrence LEFT completion n=3 partial sum after coefficient1 = 269

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[13, 64];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 13, 13], [1, 2, 4, 64, 256]]

new a_3=269;next state=[64, 269];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 64, 269]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 64)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 64

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(0, 64)

recurrence LEFT completion n=4 partial sum after coefficient0 = 64

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 269)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 1076

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(64, 1076)

recurrence LEFT completion n=4 partial sum after coefficient1 = 1140

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[64, 269];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 64, 64], [1, 3, 4, 269, 1076]]

new a_4=1140;next state=[269, 1140];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 64, 269, 1140]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 269)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 269

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(0, 269)

recurrence LEFT completion n=5 partial sum after coefficient0 = 269

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 1140)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 4560

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(269, 4560)

recurrence LEFT completion n=5 partial sum after coefficient1 = 4829

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[269, 1140];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 269, 269], [1, 4, 4, 1140, 4560]]

new a_5=4829;next state=[1140, 4829];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 64, 269, 1140, 4829]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 1140)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 1140

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(0, 1140)

recurrence LEFT completion n=6 partial sum after coefficient0 = 1140

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 4829)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 19316

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(1140, 19316)

recurrence LEFT completion n=6 partial sum after coefficient1 = 20456

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1140, 4829];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 1140, 1140], [1, 5, 4, 4829, 19316]]

new a_6=20456;next state=[4829, 20456];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 64, 269, 1140, 4829, 20456]

recurrence LEFT completion:complete ORIGINAL initial state=[12, 13];actual generated prefix=[12, 13, 64, 269, 1140, 4829, 20456]

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, 4];constant=0;ALL initial a_0..a_(r-1)=[12, 14]

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, 12)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 12

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(0, 12)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 12

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(4, 14)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 56

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(12, 56)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 68

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[12, 14];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 12, 12], [1, 1, 4, 14, 56]]

new a_2=68;next state=[14, 68];drop ONLY oldest state entry then append new scalar;prefix now=[12, 14, 68]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 14)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 14

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(0, 14)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 14

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 68)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 272

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(14, 272)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 286

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[14, 68];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 14, 14], [1, 2, 4, 68, 272]]

new a_3=286;next state=[68, 286];drop ONLY oldest state entry then append new scalar;prefix now=[12, 14, 68, 286]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 68)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 68

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(0, 68)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 68

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 286)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 1144

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(68, 1144)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 1212

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[68, 286];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 68, 68], [1, 3, 4, 286, 1144]]

new a_4=1212;next state=[286, 1212];drop ONLY oldest state entry then append new scalar;prefix now=[12, 14, 68, 286, 1212]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 286)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 286

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(0, 286)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 286

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 1212)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 4848

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(286, 4848)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 5134

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[286, 1212];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 286, 286], [1, 4, 4, 1212, 4848]]

new a_5=5134;next state=[1212, 5134];drop ONLY oldest state entry then append new scalar;prefix now=[12, 14, 68, 286, 1212, 5134]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 1212)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 1212

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(0, 1212)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 1212

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 5134)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 20536

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(1212, 20536)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 21748

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1212, 5134];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 1212, 1212], [1, 5, 4, 5134, 20536]]

new a_6=21748;next state=[5134, 21748];drop ONLY oldest state entry then append new scalar;prefix now=[12, 14, 68, 286, 1212, 5134, 21748]

recurrence RIGHT completion:complete ORIGINAL initial state=[12, 14];actual generated prefix=[12, 14, 68, 286, 1212, 5134, 21748]

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=12;different completions a_1=13 and14;target=6

WHOLE prefixes including initials differ=是;actual targets=[20456, 21748] 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

(12, 13, 14, [12, 13, 64, 269, 1140, 4829, 20456], [12, 14, 68, 286, 1212, 5134, 21748], 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。