數列是應用數學中經常出現的觀念，而遞迴關係是一種非常有效研究數列的工具。
本文將探討遞迴關係一些常見的解法及其在演算法、組合學、代數、分析、機率上的
一些重要應用，並討論一些數學競賽中與遞迴關係相關的問題。

Sequences often occur in many branches of applied mathematics. Recurrence
relation is a powerful tool to characterize and study sequences. Some
commonly used methods for solving recurrence relations will be investigated.
Many examples with applications in algorithm, combination, algebra, analysis,
probability, etc, will be discussed. Finally, some well-known contest
problems related to recurrence relations will be addressed.

致謝 i
中文摘要 ii
英文摘要 iii
第一節 前言 1
第二節 遞迴關係(recurrence relation) 2
2.1 定義 2
2.2 著名的例子 4
第三節 一階線性遞迴關係(first-order linear recurrence relation) 5
3.1 齊次一階線性遞迴關係(first-order homogeneous linear recurrence relation) 5
3.2 非齊次一階線性遞迴關係(first-order nonhomogeneous linear recurrence relation) 6
第四節 常係數線性遞迴關係(linear recurrence relation with constant coefficients) 12
4.1 齊次常係數線性遞迴關係(homogeneous linear recurrence relation with constant coefficients) 12
4.2 非齊次常係數線性遞迴關係(nonhomogeneous linear recurrence relation with constant coefficients)19
第五節 線性遞迴關係(linear recurrence relation)29
5.1 運算子法(method of operator) 29
5.2 消去法(method of annihilator) 35
5.3 變數代換法(method of change of variable) 36
第六節 非線性遞迴關係(nonlinear recurrence relation)37
5.1 各個擊破關係(method of divide-and-conquer
relation) 38
5.2 二進位表示法(method of binary number system)
39
5.3 對數轉換法(method of logarithmic
transformation) . 41
第七節 生成函數法 (method of generating function)43
第八節 數學競賽中的遞迴關係 55
附錄A：Mathematica : RSolve 61
附錄B：整數數列百科全書 (The On-Line Encyclopedia
of Integer Sequences) ..63
附錄C：習題解答提示 64
附錄D：縮寫與符號 65
D.1 縮寫 65
D.2 符號 65
附錄E：重點整理 65
E.1 2 遞迴關係 65
E.2 3 一階線性遞迴關係 66
E.3 4 常係數線性遞迴關係 66
E.4 5 線性遞迴關係 68
E.5 6 非線性遞迴關係 69
E.6 7 生成函數法 69
參考文獻 71
索引 72

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