第 22554 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":[1,5],"constant":3,"first_initial":3,"second_left":4,"second_right":5,"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":[1,5],"constant":3,"first_initial":3,"second_left":4,"second_right":5,"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, 5];constant=3;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 LEFT completion n=2 ORIGINAL coefficient0 times a_0 := mul(1, 3)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 3

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

recurrence LEFT completion n=2 partial sum after coefficient0 = 6

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(5, 4)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 20

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(6, 20)

recurrence LEFT completion n=2 partial sum after coefficient1 = 26

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[3, 4];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 3, 3], [1, 1, 5, 4, 20]]

new a_2=26;next state=[4, 26];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 26]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 4)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 4

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

recurrence LEFT completion n=3 partial sum after coefficient0 = 7

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(5, 26)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 130

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(7, 130)

recurrence LEFT completion n=3 partial sum after coefficient1 = 137

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[4, 26];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 4, 4], [1, 2, 5, 26, 130]]

new a_3=137;next state=[26, 137];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 26, 137]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 26)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 26

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(3, 26)

recurrence LEFT completion n=4 partial sum after coefficient0 = 29

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(5, 137)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 685

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(29, 685)

recurrence LEFT completion n=4 partial sum after coefficient1 = 714

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[26, 137];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 26, 26], [1, 3, 5, 137, 685]]

new a_4=714;next state=[137, 714];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 26, 137, 714]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 137)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 137

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(3, 137)

recurrence LEFT completion n=5 partial sum after coefficient0 = 140

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(5, 714)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 3570

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(140, 3570)

recurrence LEFT completion n=5 partial sum after coefficient1 = 3710

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[137, 714];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 137, 137], [1, 4, 5, 714, 3570]]

new a_5=3710;next state=[714, 3710];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 26, 137, 714, 3710]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 714)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 714

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(3, 714)

recurrence LEFT completion n=6 partial sum after coefficient0 = 717

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(5, 3710)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 18550

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(717, 18550)

recurrence LEFT completion n=6 partial sum after coefficient1 = 19267

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[714, 3710];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 714, 714], [1, 5, 5, 3710, 18550]]

new a_6=19267;next state=[3710, 19267];drop ONLY oldest state entry then append new scalar;prefix now=[3, 4, 26, 137, 714, 3710, 19267]

recurrence LEFT completion:complete ORIGINAL initial state=[3, 4];actual generated prefix=[3, 4, 26, 137, 714, 3710, 19267]

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, 5];constant=3;ALL initial a_0..a_(r-1)=[3, 5]

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, 3)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 3

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

recurrence RIGHT completion n=2 partial sum after coefficient0 = 6

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(5, 5)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 25

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(6, 25)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 31

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[3, 5];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 3, 3], [1, 1, 5, 5, 25]]

new a_2=31;next state=[5, 31];drop ONLY oldest state entry then append new scalar;prefix now=[3, 5, 31]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 5)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 5

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

recurrence RIGHT completion n=3 partial sum after coefficient0 = 8

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(5, 31)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 155

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(8, 155)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 163

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[5, 31];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 5, 5], [1, 2, 5, 31, 155]]

new a_3=163;next state=[31, 163];drop ONLY oldest state entry then append new scalar;prefix now=[3, 5, 31, 163]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 31)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 31

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(3, 31)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 34

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(5, 163)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 815

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(34, 815)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 849

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[31, 163];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 31, 31], [1, 3, 5, 163, 815]]

new a_4=849;next state=[163, 849];drop ONLY oldest state entry then append new scalar;prefix now=[3, 5, 31, 163, 849]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 163)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 163

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(3, 163)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 166

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(5, 849)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 4245

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(166, 4245)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 4411

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[163, 849];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 163, 163], [1, 4, 5, 849, 4245]]

new a_5=4411;next state=[849, 4411];drop ONLY oldest state entry then append new scalar;prefix now=[3, 5, 31, 163, 849, 4411]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 849)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 849

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(3, 849)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 852

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(5, 4411)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 22055

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(852, 22055)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 22907

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[849, 4411];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 849, 849], [1, 5, 5, 4411, 22055]]

new a_6=22907;next state=[4411, 22907];drop ONLY oldest state entry then append new scalar;prefix now=[3, 5, 31, 163, 849, 4411, 22907]

recurrence RIGHT completion:complete ORIGINAL initial state=[3, 5];actual generated prefix=[3, 5, 31, 163, 849, 4411, 22907]

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=3;different completions a_1=4 and5;target=6

WHOLE prefixes including initials differ=是;actual targets=[19267, 22907] 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

(3, 4, 5, [3, 4, 26, 137, 714, 3710, 19267], [3, 5, 31, 163, 849, 4411, 22907], 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。