第 24695 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":20,"a1":21,"x":[1,5]}。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":20,"a1":21,"x":[1,5]}。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(20, 21)

OGF complete recurrence coefficient2 = 41

OGF complete recurrence coefficient3 := sum(21, 41)

OGF complete recurrence coefficient3 = 62

OGF complete recurrence coefficient4 := sum(41, 62)

OGF complete recurrence coefficient4 = 103

OGF complete recurrence coefficient5 := sum(62, 103)

OGF complete recurrence coefficient5 = 165

OGF complete recurrence coefficient6 := sum(103, 165)

OGF complete recurrence coefficient6 = 268

OGF complete recurrence coefficient7 := sum(165, 268)

OGF complete recurrence coefficient7 = 433

OGF complete recurrence coefficient8 := sum(268, 433)

OGF complete recurrence coefficient8 = 701

OGF complete recurrence coefficient9 := sum(433, 701)

OGF complete recurrence coefficient9 = 1134

OGF initial a1 minus a0 boundary := sub(21, 20)

OGF initial a1 minus a0 boundary = 1

OGF initial boundary times x := mul(1, 1/5)

OGF initial boundary times x = 1/5

OGF numerator with BOTH initial terms := sum(20, 1/5)

OGF numerator with BOTH initial terms = 101/5

(1/5)^2

OGF original x square = 1/25

OGF denominator one minus x := sub(1, 1/5)

OGF denominator one minus x = 4/5

OGF denominator subtract x squared := sub(4/5, 1/25)

OGF denominator subtract x squared = 19/25

OGF recurrence analytic closed value := div(101/5, 19/25)

OGF recurrence analytic closed value = 505/19

source roles = [20, 21, 1/5, [20, 21, 41, 62, 103, 165, 268, 433, 701, 1134]]

derived = [101/5, 19/25, 505/19, (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

(20, 21, [20, 21, 41, 62, 103, 165, 268, 433, 701, 1134], 1/5, 101/5, 19/25, 505/19, 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。