Goals of This Book

Sequences, both finite and infinite, are ubiquitous in mathematics and theoretical computer science. Sloane and Plouffe's book, The Encyclopedia of Integer Sequences, lists over 5,000 interesting sequences from the mathematical literature.

Sloane's web site,

http://www.research.att.com/~njas/sequences/index.html gives access to more than 69,000 sequences. There is a web-based scholarly journal, the Journal of Integer Sequences, devoted to sequence-related topics, and even a periodic international conference, SETA (Sequences and Their Applications), devoted to the study of sequences.

Sequences come in all flavors. Some, such as periodic sequences, are highly ordered and very easy to describe, while others, such as random sequences, are unordered and have no simple description.

The subject of this book is automatic sequences and their generalizations. Automatic sequences form a class of sequences somewhere between simple order and chaotic disorder. This class contains such celebrated sequences as the Thue—Morse sequence (see Chapters 1 and 6) and the Rudin—Shapiro sequence (see Chapter 3), which play important roles in many different areas of mathematics.

Automatic sequences are generated by finite automata, one of the most basic models of computation. Finite automata and other computational models are introduced in Chapter 5. Automatic sequences are also generated by iterating a simple kind of map, called a uniform morphism; see Chapter 6.