第 24682 題：解析普通生成函數

原題與已知條件

A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={"a0":7,"a1":8,"x":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-recurrence-boundary-terms；資料={"a0":7,"a1":8,"x":[1,4]}。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-analytic-ordinary-generating-function

逐步推導

source roles = [ogf-recurrence-boundary-terms]

derived = [[a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate]]

OGF complete recurrence coefficient2 := sum(7, 8)

OGF complete recurrence coefficient2 = 15

OGF complete recurrence coefficient3 := sum(8, 15)

OGF complete recurrence coefficient3 = 23

OGF complete recurrence coefficient4 := sum(15, 23)

OGF complete recurrence coefficient4 = 38

OGF complete recurrence coefficient5 := sum(23, 38)

OGF complete recurrence coefficient5 = 61

OGF complete recurrence coefficient6 := sum(38, 61)

OGF complete recurrence coefficient6 = 99

OGF complete recurrence coefficient7 := sum(61, 99)

OGF complete recurrence coefficient7 = 160

OGF complete recurrence coefficient8 := sum(99, 160)

OGF complete recurrence coefficient8 = 259

OGF complete recurrence coefficient9 := sum(160, 259)

OGF complete recurrence coefficient9 = 419

OGF initial a1 minus a0 boundary := sub(8, 7)

OGF initial a1 minus a0 boundary = 1

OGF initial boundary times x := mul(1, 1/4)

OGF initial boundary times x = 1/4

OGF numerator with BOTH initial terms := sum(7, 1/4)

OGF numerator with BOTH initial terms = 29/4

(1/4)^2

OGF original x square = 1/16

OGF denominator one minus x := sub(1, 1/4)

OGF denominator one minus x = 3/4

OGF denominator subtract x squared := sub(3/4, 1/16)

OGF denominator subtract x squared = 11/16

OGF recurrence analytic closed value := div(29/4, 11/16)

OGF recurrence analytic closed value = 116/11

source roles = [7, 8, 1/4, [7, 8, 15, 23, 38, 61, 99, 160, 259, 419]]

derived = [29/4, 11/16, 116/11, (1-x-x^2)A=a0+(a1-a0)x]

逐步反向核對

• 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 OGF recurrence analytic closed value，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 OGF denominator subtract x squared，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 OGF denominator one minus x，不以小數近似替代。
• S013 · 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。
• S012 · premise_and_result_audit — 以精確有理數重算 OGF numerator with BOTH initial terms，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 OGF initial boundary times x，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 OGF initial a1 minus a0 boundary，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 OGF complete recurrence coefficient9，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 OGF complete recurrence coefficient8，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 OGF complete recurrence coefficient7，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 OGF complete recurrence coefficient6，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 OGF complete recurrence coefficient5，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 OGF complete recurrence coefficient4，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 OGF complete recurrence coefficient3，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 OGF complete recurrence coefficient2，不以小數近似替代。
• 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

(7, 8, [7, 8, 15, 23, 38, 61, 99, 160, 259, 419], 1/4, 29/4, 11/16, 116/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)

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。