第 24693 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":18,"a1":19,"x":[1,6]}。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":18,"a1":19,"x":[1,6]}。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-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(18, 19)

OGF complete recurrence coefficient2 = 37

OGF complete recurrence coefficient3 := sum(19, 37)

OGF complete recurrence coefficient3 = 56

OGF complete recurrence coefficient4 := sum(37, 56)

OGF complete recurrence coefficient4 = 93

OGF complete recurrence coefficient5 := sum(56, 93)

OGF complete recurrence coefficient5 = 149

OGF complete recurrence coefficient6 := sum(93, 149)

OGF complete recurrence coefficient6 = 242

OGF complete recurrence coefficient7 := sum(149, 242)

OGF complete recurrence coefficient7 = 391

OGF complete recurrence coefficient8 := sum(242, 391)

OGF complete recurrence coefficient8 = 633

OGF complete recurrence coefficient9 := sum(391, 633)

OGF complete recurrence coefficient9 = 1024

OGF initial a1 minus a0 boundary := sub(19, 18)

OGF initial a1 minus a0 boundary = 1

OGF initial boundary times x := mul(1, 1/6)

OGF initial boundary times x = 1/6

OGF numerator with BOTH initial terms := sum(18, 1/6)

OGF numerator with BOTH initial terms = 109/6

(1/6)^2

OGF original x square = 1/36

OGF denominator one minus x := sub(1, 1/6)

OGF denominator one minus x = 5/6

OGF denominator subtract x squared := sub(5/6, 1/36)

OGF denominator subtract x squared = 29/36

OGF recurrence analytic closed value := div(109/6, 29/36)

OGF recurrence analytic closed value = 654/29

source roles = [18, 19, 1/6, [18, 19, 37, 56, 93, 149, 242, 391, 633, 1024]]

derived = [109/6, 29/36, 654/29, (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

(18, 19, [18, 19, 37, 56, 93, 149, 242, 391, 633, 1024], 1/6, 109/6, 29/36, 654/29, 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。