# 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
4.9	Phi and Mu
Exercises

5.  Binomial Coefficients
5.1	Basic Identities
5.2	Basic Practice
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	0 Notation
9.3	0 Manipulation
9.4	Two Asymptotic Tricks
9.5	Buler's Summation Formula
9.6	Final Summations
Exercises