第 22551 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":[1,2],"constant":0,"first_initial":0,"second_left":1,"second_right":2,"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":[1,2],"constant":0,"first_initial":0,"second_left":1,"second_right":2,"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, 2];constant=0;ALL initial a_0..a_(r-1)=[0, 1]

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, 0)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 0

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

recurrence LEFT completion n=2 partial sum after coefficient0 = 0

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(2, 1)

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

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(0, 2)

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

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

new a_2=2;next state=[1, 2];drop ONLY oldest state entry then append new scalar;prefix now=[0, 1, 2]

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

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 1

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(0, 1)

recurrence LEFT completion n=3 partial sum after coefficient0 = 1

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(2, 2)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 4

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(1, 4)

recurrence LEFT completion n=3 partial sum after coefficient1 = 5

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

new a_3=5;next state=[2, 5];drop ONLY oldest state entry then append new scalar;prefix now=[0, 1, 2, 5]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 2)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 2

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(0, 2)

recurrence LEFT completion n=4 partial sum after coefficient0 = 2

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(2, 5)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 10

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(2, 10)

recurrence LEFT completion n=4 partial sum after coefficient1 = 12

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[2, 5];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 2, 2], [1, 3, 2, 5, 10]]

new a_4=12;next state=[5, 12];drop ONLY oldest state entry then append new scalar;prefix now=[0, 1, 2, 5, 12]

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

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

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

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

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(2, 12)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 24

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(5, 24)

recurrence LEFT completion n=5 partial sum after coefficient1 = 29

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[5, 12];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 5, 5], [1, 4, 2, 12, 24]]

new a_5=29;next state=[12, 29];drop ONLY oldest state entry then append new scalar;prefix now=[0, 1, 2, 5, 12, 29]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 12)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 12

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(0, 12)

recurrence LEFT completion n=6 partial sum after coefficient0 = 12

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(2, 29)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 58

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(12, 58)

recurrence LEFT completion n=6 partial sum after coefficient1 = 70

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[12, 29];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 12, 12], [1, 5, 2, 29, 58]]

new a_6=70;next state=[29, 70];drop ONLY oldest state entry then append new scalar;prefix now=[0, 1, 2, 5, 12, 29, 70]

recurrence LEFT completion:complete ORIGINAL initial state=[0, 1];actual generated prefix=[0, 1, 2, 5, 12, 29, 70]

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, 2];constant=0;ALL initial a_0..a_(r-1)=[0, 2]

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, 0)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 0

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

recurrence RIGHT completion n=2 partial sum after coefficient0 = 0

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

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 4

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(0, 4)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 4

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

new a_2=4;next state=[2, 4];drop ONLY oldest state entry then append new scalar;prefix now=[0, 2, 4]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 2)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 2

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(0, 2)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 2

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(2, 4)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 8

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(2, 8)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 10

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

new a_3=10;next state=[4, 10];drop ONLY oldest state entry then append new scalar;prefix now=[0, 2, 4, 10]

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

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

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

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

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(2, 10)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 20

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(4, 20)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 24

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[4, 10];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 4, 4], [1, 3, 2, 10, 20]]

new a_4=24;next state=[10, 24];drop ONLY oldest state entry then append new scalar;prefix now=[0, 2, 4, 10, 24]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 10)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 10

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(0, 10)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 10

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(2, 24)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 48

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(10, 48)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 58

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[10, 24];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 10, 10], [1, 4, 2, 24, 48]]

new a_5=58;next state=[24, 58];drop ONLY oldest state entry then append new scalar;prefix now=[0, 2, 4, 10, 24, 58]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 24)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 24

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(0, 24)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 24

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(2, 58)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 116

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(24, 116)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 140

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[24, 58];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 24, 24], [1, 5, 2, 58, 116]]

new a_6=140;next state=[58, 140];drop ONLY oldest state entry then append new scalar;prefix now=[0, 2, 4, 10, 24, 58, 140]

recurrence RIGHT completion:complete ORIGINAL initial state=[0, 2];actual generated prefix=[0, 2, 4, 10, 24, 58, 140]

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=0;different completions a_1=1 and2;target=6

WHOLE prefixes including initials differ=是;actual targets=[70, 140] 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

(0, 1, 2, [0, 1, 2, 5, 12, 29, 70], [0, 2, 4, 10, 24, 58, 140], 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。