第 22566 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":3,"first_initial":15,"second_left":16,"second_right":17,"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":3,"first_initial":15,"second_left":16,"second_right":17,"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=3;ALL initial a_0..a_(r-1)=[15, 16]

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, 15)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 15

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(3, 15)

recurrence LEFT completion n=2 partial sum after coefficient0 = 18

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(2, 16)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 32

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(18, 32)

recurrence LEFT completion n=2 partial sum after coefficient1 = 50

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[15, 16];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 15, 15], [1, 1, 2, 16, 32]]

new a_2=50;next state=[16, 50];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 50]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 16)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 16

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(3, 16)

recurrence LEFT completion n=3 partial sum after coefficient0 = 19

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(2, 50)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 100

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(19, 100)

recurrence LEFT completion n=3 partial sum after coefficient1 = 119

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[16, 50];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 16, 16], [1, 2, 2, 50, 100]]

new a_3=119;next state=[50, 119];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 50, 119]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 50)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 50

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(3, 50)

recurrence LEFT completion n=4 partial sum after coefficient0 = 53

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(2, 119)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 238

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(53, 238)

recurrence LEFT completion n=4 partial sum after coefficient1 = 291

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[50, 119];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 50, 50], [1, 3, 2, 119, 238]]

new a_4=291;next state=[119, 291];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 50, 119, 291]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 119)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 119

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(3, 119)

recurrence LEFT completion n=5 partial sum after coefficient0 = 122

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(2, 291)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 582

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(122, 582)

recurrence LEFT completion n=5 partial sum after coefficient1 = 704

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[119, 291];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 119, 119], [1, 4, 2, 291, 582]]

new a_5=704;next state=[291, 704];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 50, 119, 291, 704]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 291)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 291

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(3, 291)

recurrence LEFT completion n=6 partial sum after coefficient0 = 294

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(2, 704)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 1408

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(294, 1408)

recurrence LEFT completion n=6 partial sum after coefficient1 = 1702

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[291, 704];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 291, 291], [1, 5, 2, 704, 1408]]

new a_6=1702;next state=[704, 1702];drop ONLY oldest state entry then append new scalar;prefix now=[15, 16, 50, 119, 291, 704, 1702]

recurrence LEFT completion:complete ORIGINAL initial state=[15, 16];actual generated prefix=[15, 16, 50, 119, 291, 704, 1702]

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=3;ALL initial a_0..a_(r-1)=[15, 17]

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, 15)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 15

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(3, 15)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 18

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(2, 17)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 34

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(18, 34)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 52

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[15, 17];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 15, 15], [1, 1, 2, 17, 34]]

new a_2=52;next state=[17, 52];drop ONLY oldest state entry then append new scalar;prefix now=[15, 17, 52]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 17)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 17

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(3, 17)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 20

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(2, 52)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 104

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(20, 104)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 124

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[17, 52];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 17, 17], [1, 2, 2, 52, 104]]

new a_3=124;next state=[52, 124];drop ONLY oldest state entry then append new scalar;prefix now=[15, 17, 52, 124]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 52)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 52

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(3, 52)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 55

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(2, 124)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 248

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(55, 248)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 303

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[52, 124];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 52, 52], [1, 3, 2, 124, 248]]

new a_4=303;next state=[124, 303];drop ONLY oldest state entry then append new scalar;prefix now=[15, 17, 52, 124, 303]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 124)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 124

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(3, 124)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 127

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(2, 303)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 606

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(127, 606)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 733

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[124, 303];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 124, 124], [1, 4, 2, 303, 606]]

new a_5=733;next state=[303, 733];drop ONLY oldest state entry then append new scalar;prefix now=[15, 17, 52, 124, 303, 733]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 303)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 303

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(3, 303)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 306

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(2, 733)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 1466

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(306, 1466)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 1772

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[303, 733];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 303, 303], [1, 5, 2, 733, 1466]]

new a_6=1772;next state=[733, 1772];drop ONLY oldest state entry then append new scalar;prefix now=[15, 17, 52, 124, 303, 733, 1772]

recurrence RIGHT completion:complete ORIGINAL initial state=[15, 17];actual generated prefix=[15, 17, 52, 124, 303, 733, 1772]

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=15;different completions a_1=16 and17;target=6

WHOLE prefixes including initials differ=是;actual targets=[1702, 1772] 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

(15, 16, 17, [15, 16, 50, 119, 291, 704, 1702], [15, 17, 52, 124, 303, 733, 1772], 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。