## Appendix C

## Solutions of Recurrence Relations

#### C.1 Introduction

A recurrence relation for a sequence is an equation that defines the present term by previous terms.

Example: *a*_{n} = *a*_{n} _{– 1} + *a*_{n} _{– 2}

The **initial conditions** specify the terms before the recurrence relation takes effect.

Example: *a*_{1} = 2; *a*_{2} = 3

Many counting problems can be modelled by recurrence relations.

**Example 1**

How many bit strings of length *n* have no “00”? All strings begin with a 1 or a 0. There are total 2^{n} strings of length *n*. Out of these, we accept the following strings:

- Starts with a 1; remaining
*n* – 1; bits can be any string with no 00.
- Starts with a 0 and 2nd bit is a 1; remaining
*n* – 2 bits can be any string with no 00.
- 2 strings of length 1 and 3 strings of length ...