第 22553 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":[3,4],"constant":2,"first_initial":2,"second_left":3,"second_right":4,"target_n":8}。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":[3,4],"constant":2,"first_initial":2,"second_left":3,"second_right":4,"target_n":8}。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=[3, 4];constant=2;ALL initial a_0..a_(r-1)=[2, 3]

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(3, 2)

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(4, 3)

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 12

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(8, 12)

recurrence LEFT completion n=2 partial sum after coefficient1 = 20

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[2, 3];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 2, 6], [1, 1, 4, 3, 12]]

new a_2=20;next state=[3, 20];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20]

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

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 9

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(2, 9)

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

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 20)

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 80

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(11, 80)

recurrence LEFT completion n=3 partial sum after coefficient1 = 91

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[3, 20];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 3, 9], [1, 2, 4, 20, 80]]

new a_3=91;next state=[20, 91];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91]

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 20)

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 60

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(2, 60)

recurrence LEFT completion n=4 partial sum after coefficient0 = 62

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 91)

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 364

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(62, 364)

recurrence LEFT completion n=4 partial sum after coefficient1 = 426

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[20, 91];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 20, 60], [1, 3, 4, 91, 364]]

new a_4=426;next state=[91, 426];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91, 426]

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 91)

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 273

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(2, 273)

recurrence LEFT completion n=5 partial sum after coefficient0 = 275

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 426)

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 1704

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(275, 1704)

recurrence LEFT completion n=5 partial sum after coefficient1 = 1979

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[91, 426];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 91, 273], [1, 4, 4, 426, 1704]]

new a_5=1979;next state=[426, 1979];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91, 426, 1979]

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 426)

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 1278

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(2, 1278)

recurrence LEFT completion n=6 partial sum after coefficient0 = 1280

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 1979)

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 7916

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(1280, 7916)

recurrence LEFT completion n=6 partial sum after coefficient1 = 9196

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[426, 1979];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 426, 1278], [1, 5, 4, 1979, 7916]]

new a_6=9196;next state=[1979, 9196];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91, 426, 1979, 9196]

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 1979)

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 5937

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(2, 5937)

recurrence LEFT completion n=7 partial sum after coefficient0 = 5939

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(4, 9196)

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 36784

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(5939, 36784)

recurrence LEFT completion n=7 partial sum after coefficient1 = 42723

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[1979, 9196];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 1979, 5937], [1, 6, 4, 9196, 36784]]

new a_7=42723;next state=[9196, 42723];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91, 426, 1979, 9196, 42723]

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 9196)

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 = 27588

recurrence LEFT completion n=8 partial sum after coefficient0 := sum(2, 27588)

recurrence LEFT completion n=8 partial sum after coefficient0 = 27590

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 := mul(4, 42723)

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 = 170892

recurrence LEFT completion n=8 partial sum after coefficient1 := sum(27590, 170892)

recurrence LEFT completion n=8 partial sum after coefficient1 = 198482

recurrence LEFT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[9196, 42723];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 9196, 27588], [1, 7, 4, 42723, 170892]]

new a_8=198482;next state=[42723, 198482];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91, 426, 1979, 9196, 42723, 198482]

recurrence LEFT completion:complete ORIGINAL initial state=[2, 3];actual generated prefix=[2, 3, 20, 91, 426, 1979, 9196, 42723, 198482]

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=[3, 4];constant=2;ALL initial a_0..a_(r-1)=[2, 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 RIGHT completion n=2 ORIGINAL coefficient0 times a_0 := mul(3, 2)

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(4, 4)

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 16

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(8, 16)

recurrence RIGHT completion n=2 partial sum after coefficient1 = 24

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[2, 4];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 2, 6], [1, 1, 4, 4, 16]]

new a_2=24;next state=[4, 24];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24]

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

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 12

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(2, 12)

recurrence RIGHT completion n=3 partial sum after coefficient0 = 14

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 24)

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

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(14, 96)

recurrence RIGHT completion n=3 partial sum after coefficient1 = 110

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[4, 24];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 4, 12], [1, 2, 4, 24, 96]]

new a_3=110;next state=[24, 110];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110]

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 24)

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 72

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(2, 72)

recurrence RIGHT completion n=4 partial sum after coefficient0 = 74

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 110)

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 440

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(74, 440)

recurrence RIGHT completion n=4 partial sum after coefficient1 = 514

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[24, 110];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 24, 72], [1, 3, 4, 110, 440]]

new a_4=514;next state=[110, 514];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110, 514]

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 110)

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 330

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(2, 330)

recurrence RIGHT completion n=5 partial sum after coefficient0 = 332

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 514)

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 2056

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(332, 2056)

recurrence RIGHT completion n=5 partial sum after coefficient1 = 2388

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[110, 514];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 110, 330], [1, 4, 4, 514, 2056]]

new a_5=2388;next state=[514, 2388];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110, 514, 2388]

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 514)

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 1542

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(2, 1542)

recurrence RIGHT completion n=6 partial sum after coefficient0 = 1544

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 2388)

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 9552

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(1544, 9552)

recurrence RIGHT completion n=6 partial sum after coefficient1 = 11096

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[514, 2388];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 514, 1542], [1, 5, 4, 2388, 9552]]

new a_6=11096;next state=[2388, 11096];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110, 514, 2388, 11096]

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 2388)

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 7164

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(2, 7164)

recurrence RIGHT completion n=7 partial sum after coefficient0 = 7166

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(4, 11096)

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 44384

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(7166, 44384)

recurrence RIGHT completion n=7 partial sum after coefficient1 = 51550

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[2388, 11096];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 2388, 7164], [1, 6, 4, 11096, 44384]]

new a_7=51550;next state=[11096, 51550];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110, 514, 2388, 11096, 51550]

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 11096)

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 = 33288

recurrence RIGHT completion n=8 partial sum after coefficient0 := sum(2, 33288)

recurrence RIGHT completion n=8 partial sum after coefficient0 = 33290

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 := mul(4, 51550)

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 = 206200

recurrence RIGHT completion n=8 partial sum after coefficient1 := sum(33290, 206200)

recurrence RIGHT completion n=8 partial sum after coefficient1 = 239490

recurrence RIGHT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[11096, 51550];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 11096, 33288], [1, 7, 4, 51550, 206200]]

new a_8=239490;next state=[51550, 239490];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110, 514, 2388, 11096, 51550, 239490]

recurrence RIGHT completion:complete ORIGINAL initial state=[2, 4];actual generated prefix=[2, 4, 24, 110, 514, 2388, 11096, 51550, 239490]

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=2;different completions a_1=3 and4;target=8

WHOLE prefixes including initials differ=是;actual targets=[198482, 239490] MAY coincide;one_initial_uniqueness_claim fixed No

逐步反向核對

• S076 · 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。
• S075 · 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。
• S074 · 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。
• S073 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=8 partial sum after coefficient1，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=8 partial sum after coefficient0，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6，不以小數近似替代。
• S069 · 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。
• S068 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=7 partial sum after coefficient1，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=7 partial sum after coefficient0，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5，不以小數近似替代。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · 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。
• S057 · 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。
• S056 · 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。
• S055 · 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。
• 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=4 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=4 ORIGINAL coefficient1 times a_3，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=4 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=4 ORIGINAL coefficient0 times a_2，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=3 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=3 ORIGINAL coefficient1 times a_2，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=3 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=3 ORIGINAL coefficient0 times a_1，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=2 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=2 ORIGINAL coefficient1 times a_1，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=2 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=2 ORIGINAL coefficient0 times a_0，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S037 · 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。
• S036 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=8 partial sum after coefficient1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=8 partial sum after coefficient0，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6，不以小數近似替代。
• S032 · 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。
• S031 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=7 partial sum after coefficient1，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=7 partial sum after coefficient0，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5，不以小數近似替代。
• 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

(2, 3, 4, [2, 3, 20, 91, 426, 1979, 9196, 42723, 198482], [2, 4, 24, 110, 514, 2388, 11096, 51550, 239490], 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。