第 22557 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":[1,3],"constant":2,"first_initial":6,"second_left":7,"second_right":8,"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":[1,3],"constant":2,"first_initial":6,"second_left":7,"second_right":8,"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, 3];constant=2;ALL initial a_0..a_(r-1)=[6, 7]

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, 6)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 6

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(2, 6)

recurrence LEFT completion n=2 partial sum after coefficient0 = 8

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 7)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 21

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(8, 21)

recurrence LEFT completion n=2 partial sum after coefficient1 = 29

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[6, 7];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 6, 6], [1, 1, 3, 7, 21]]

new a_2=29;next state=[7, 29];drop ONLY oldest state entry then append new scalar;prefix now=[6, 7, 29]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 7)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 7

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(2, 7)

recurrence LEFT completion n=3 partial sum after coefficient0 = 9

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 29)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 87

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(9, 87)

recurrence LEFT completion n=3 partial sum after coefficient1 = 96

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[7, 29];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 7, 7], [1, 2, 3, 29, 87]]

new a_3=96;next state=[29, 96];drop ONLY oldest state entry then append new scalar;prefix now=[6, 7, 29, 96]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 29)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 29

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(2, 29)

recurrence LEFT completion n=4 partial sum after coefficient0 = 31

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 96)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 288

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(31, 288)

recurrence LEFT completion n=4 partial sum after coefficient1 = 319

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[29, 96];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 29, 29], [1, 3, 3, 96, 288]]

new a_4=319;next state=[96, 319];drop ONLY oldest state entry then append new scalar;prefix now=[6, 7, 29, 96, 319]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 96)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 96

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(2, 96)

recurrence LEFT completion n=5 partial sum after coefficient0 = 98

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 319)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 957

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(98, 957)

recurrence LEFT completion n=5 partial sum after coefficient1 = 1055

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[96, 319];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 96, 96], [1, 4, 3, 319, 957]]

new a_5=1055;next state=[319, 1055];drop ONLY oldest state entry then append new scalar;prefix now=[6, 7, 29, 96, 319, 1055]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 319)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 319

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(2, 319)

recurrence LEFT completion n=6 partial sum after coefficient0 = 321

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 1055)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 3165

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(321, 3165)

recurrence LEFT completion n=6 partial sum after coefficient1 = 3486

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[319, 1055];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 319, 319], [1, 5, 3, 1055, 3165]]

new a_6=3486;next state=[1055, 3486];drop ONLY oldest state entry then append new scalar;prefix now=[6, 7, 29, 96, 319, 1055, 3486]

recurrence LEFT completion:complete ORIGINAL initial state=[6, 7];actual generated prefix=[6, 7, 29, 96, 319, 1055, 3486]

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, 3];constant=2;ALL initial a_0..a_(r-1)=[6, 8]

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, 6)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 6

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(2, 6)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 8

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 8)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 24

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(8, 24)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 32

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[6, 8];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 6, 6], [1, 1, 3, 8, 24]]

new a_2=32;next state=[8, 32];drop ONLY oldest state entry then append new scalar;prefix now=[6, 8, 32]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 8)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 8

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(2, 8)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 10

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 32)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 96

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(10, 96)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 106

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[8, 32];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 8, 8], [1, 2, 3, 32, 96]]

new a_3=106;next state=[32, 106];drop ONLY oldest state entry then append new scalar;prefix now=[6, 8, 32, 106]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 32)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 32

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(2, 32)

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

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 106)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 318

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(34, 318)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 352

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[32, 106];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 32, 32], [1, 3, 3, 106, 318]]

new a_4=352;next state=[106, 352];drop ONLY oldest state entry then append new scalar;prefix now=[6, 8, 32, 106, 352]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 106)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 106

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(2, 106)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 108

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 352)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 1056

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(108, 1056)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 1164

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[106, 352];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 106, 106], [1, 4, 3, 352, 1056]]

new a_5=1164;next state=[352, 1164];drop ONLY oldest state entry then append new scalar;prefix now=[6, 8, 32, 106, 352, 1164]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 352)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 352

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(2, 352)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 354

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 1164)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 3492

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(354, 3492)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 3846

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[352, 1164];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 352, 352], [1, 5, 3, 1164, 3492]]

new a_6=3846;next state=[1164, 3846];drop ONLY oldest state entry then append new scalar;prefix now=[6, 8, 32, 106, 352, 1164, 3846]

recurrence RIGHT completion:complete ORIGINAL initial state=[6, 8];actual generated prefix=[6, 8, 32, 106, 352, 1164, 3846]

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=6;different completions a_1=7 and8;target=6

WHOLE prefixes including initials differ=是;actual targets=[3486, 3846] 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

(6, 7, 8, [6, 7, 29, 96, 319, 1055, 3486], [6, 8, 32, 106, 352, 1164, 3846], 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。