Contents

1 Recurrent Problems

1.1 The Tower of Hanoi

1.2 Lines in the Plane

1.3 The Josephus Problem

Exercises

2 Sums

2.1 Notation

2.2 Sums and Recurrences

2.3 Manipulation of Sums

2.4 Multiple Sums

2.5 General Methods

2.6 Finite and Infinite Calculus

2.7 Infinite Sums

Exercises

3 Integer Functions

3.1 Floors and Ceilings

3.2 Floor/Ceiling Applications

3.3 Floor/Ceiling Recurrences

3.4 ‘mod’: The Binary Operation

3.5 Floor/Ceiling Sums

Exercises

4 Number Theory

4.1 Divisibility

4.2 Primes

4.3 Prime Examples

4.4 Factorial Factors

4.5 Relative Primality

4.6 ‘mod’: The Congruence Relation

4.7 Independent Residues

4.8 Additional Applications

4.9 Phi and Mu

Exercises

5 Binomial Coefficients

5.1 Basic Identities

5.2 Basic Practice

5.3 Tricks of the Trade

5.4 Generating Functions

5.5 Hypergeometric Functions

5.6 Hypergeometric Transformations

5.7 Partial Hypergeometric Sums

5.8 Mechanical Summation

Exercises

6 Special Numbers

6.1 Stirling Numbers

6.2 Eulerian Numbers

6.3 Harmonic Numbers

6.4 Harmonic Summation

6.5 Bernoulli Numbers

6.6 Fibonacci Numbers

6.7 Continuants

Exercises

7 Generating Functions

7.1 Domino Theory and Change

7.2 Basic Maneuvers

7.3 Solving Recurrences

7.4 Special Generating Functions

7.5 Convolutions

7.6 Exponential Generating Functions

7.7 Dirichlet Generating Functions

Exercises

8 Discrete Probability

8.1 Definitions

8.2 Mean and Variance

8.3 Probability Generating Functions

8.4 Flipping Coins

8.5 Hashing

Exercises

9 Asymptotics

9.1 A Hierarchy

9.2 O Notation

9.3 O Manipulation

9.4 Two Asymptotic Tricks

9.5 Euler’s Summation Formula

9.6 Final Summations

Exercises

A Answers to Exercises

B Bibliography

C Credits for Exercises

Index

List of Tables

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