## 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: an = an – 1 + an – 2

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

Example: a1 = 2; a2 = 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 2n strings of length n. Out of these, we accept the following strings:

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

Get Design and Analysis of Algorithms, 2nd Edition by Pearson now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.