第 22560 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":[1,6],"constant":1,"first_initial":9,"second_left":10,"second_right":11,"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":[1,6],"constant":1,"first_initial":9,"second_left":10,"second_right":11,"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, 6];constant=1;ALL initial a_0..a_(r-1)=[9, 10]

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, 9)

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 9

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(1, 9)

recurrence LEFT completion n=2 partial sum after coefficient0 = 10

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(6, 10)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 60

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(10, 60)

recurrence LEFT completion n=2 partial sum after coefficient1 = 70

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[9, 10];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 9, 9], [1, 1, 6, 10, 60]]

new a_2=70;next state=[10, 70];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 70]

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 10)

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 10

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(1, 10)

recurrence LEFT completion n=3 partial sum after coefficient0 = 11

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(6, 70)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 420

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(11, 420)

recurrence LEFT completion n=3 partial sum after coefficient1 = 431

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[10, 70];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 10, 10], [1, 2, 6, 70, 420]]

new a_3=431;next state=[70, 431];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 70, 431]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 70)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 70

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(1, 70)

recurrence LEFT completion n=4 partial sum after coefficient0 = 71

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(6, 431)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 2586

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(71, 2586)

recurrence LEFT completion n=4 partial sum after coefficient1 = 2657

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[70, 431];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 70, 70], [1, 3, 6, 431, 2586]]

new a_4=2657;next state=[431, 2657];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 70, 431, 2657]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 431)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 431

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(1, 431)

recurrence LEFT completion n=5 partial sum after coefficient0 = 432

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(6, 2657)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 15942

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(432, 15942)

recurrence LEFT completion n=5 partial sum after coefficient1 = 16374

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[431, 2657];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 431, 431], [1, 4, 6, 2657, 15942]]

new a_5=16374;next state=[2657, 16374];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 70, 431, 2657, 16374]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 2657)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 2657

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(1, 2657)

recurrence LEFT completion n=6 partial sum after coefficient0 = 2658

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(6, 16374)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 98244

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(2658, 98244)

recurrence LEFT completion n=6 partial sum after coefficient1 = 100902

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[2657, 16374];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 2657, 2657], [1, 5, 6, 16374, 98244]]

new a_6=100902;next state=[16374, 100902];drop ONLY oldest state entry then append new scalar;prefix now=[9, 10, 70, 431, 2657, 16374, 100902]

recurrence LEFT completion:complete ORIGINAL initial state=[9, 10];actual generated prefix=[9, 10, 70, 431, 2657, 16374, 100902]

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, 6];constant=1;ALL initial a_0..a_(r-1)=[9, 11]

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, 9)

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 9

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(1, 9)

recurrence RIGHT completion n=2 partial sum after coefficient0 = 10

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(6, 11)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 66

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(10, 66)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 76

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[9, 11];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 9, 9], [1, 1, 6, 11, 66]]

new a_2=76;next state=[11, 76];drop ONLY oldest state entry then append new scalar;prefix now=[9, 11, 76]

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(1, 11)

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 11

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(1, 11)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 12

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(6, 76)

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 456

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(12, 456)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 468

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[11, 76];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 11, 11], [1, 2, 6, 76, 456]]

new a_3=468;next state=[76, 468];drop ONLY oldest state entry then append new scalar;prefix now=[9, 11, 76, 468]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(1, 76)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 76

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(1, 76)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 77

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(6, 468)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 2808

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(77, 2808)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 2885

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[76, 468];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 76, 76], [1, 3, 6, 468, 2808]]

new a_4=2885;next state=[468, 2885];drop ONLY oldest state entry then append new scalar;prefix now=[9, 11, 76, 468, 2885]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(1, 468)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 468

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(1, 468)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 469

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(6, 2885)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 17310

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(469, 17310)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 17779

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[468, 2885];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 468, 468], [1, 4, 6, 2885, 17310]]

new a_5=17779;next state=[2885, 17779];drop ONLY oldest state entry then append new scalar;prefix now=[9, 11, 76, 468, 2885, 17779]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(1, 2885)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 2885

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(1, 2885)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 2886

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(6, 17779)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 106674

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(2886, 106674)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 109560

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[2885, 17779];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 2885, 2885], [1, 5, 6, 17779, 106674]]

new a_6=109560;next state=[17779, 109560];drop ONLY oldest state entry then append new scalar;prefix now=[9, 11, 76, 468, 2885, 17779, 109560]

recurrence RIGHT completion:complete ORIGINAL initial state=[9, 11];actual generated prefix=[9, 11, 76, 468, 2885, 17779, 109560]

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=9;different completions a_1=10 and11;target=6

WHOLE prefixes including initials differ=是;actual targets=[100902, 109560] 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

(9, 10, 11, [9, 10, 70, 431, 2657, 16374, 100902], [9, 11, 76, 468, 2885, 17779, 109560], 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。