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Fibonacci and Catalan Numbers: An Introduction

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This book is in the MAA's basic library list.

MAA Review

The book is a comprehensive introduction to the Fibonacci and Catalan numbers and their many properties and uses. It is a tastefully written and well organized textbook that could be used for self study and easy reference. The book consists of two parts. The first seventeen chapters cover the Fibonacci and Lucas numbers, followed by 19 chapters that cover the Catalan numbers. Both parts touch on generalizations: the alternate Fibonacci, Narayana, Motzkin, Schröder, and Generalized Catalan numbers.

The two parts have a similar structure. A short historical background, the traditional motivating problems, additional examples that lead to the numbers under discussion, more theory and more examples, generalizations, a final example, bibliography. The index at the end of the book is common to the two parts as is the collection of solutions to the odd-numbered exercises. (Each chapter ends with a good many of those.)

The chapters “A Final Example?” come with a question mark. Why? After the many examples where the number families arise, the reader may tend to believe that if the first few terms of a sequence appear in a certain situation the rest will automatically follow. The two chapters serve a gentle warning that this is not always so. Inductive generalizations are no substitute for proof.

Naturally, there are plenty of proofs here that explore various techniques: mathematical induction, matrix algebra, recurrence relations, combinatorial arguments. The proofs are truly polished, with no omitted steps (that I could find), and should be followed easily by anybody with undergraduate math proficiency. The book grew out of a number of minicourses presented over a period of 20 years. As the author writes in the preface that “…the presentations were developed so that everyone in the audience would be able to understand at least some, if not a substantial amount, of the material.” And later on: “Since the book is to be regarded as an introduction, examples and, especially, proofs are presented with detailed explanations. Such examples and proofs are designed to be careful and thorough.”

I believe the author has achieved those goals. A casual reader — even after quick browse through the book — will see how ubiquitous the Fibonacci and Catalan numbers are and how broad are their applications in mathematics and natural sciences. A student who takes the study more seriously will, in addition, learn of their numerous and often beautiful and surprising properties, and master many methods of mathematical proof, even if the book is not read sequentially.

Alex Bogomolny is a semi-retired mathematician and web developer who likes to reminisce as to how, while writing his PhD thesis at the Hebrew University, he had to run his programs on a CDC-6400 at night because they required more than 120 KB of memory. He wonders at how the world changed in the past 30 years and looks forward to observing the progress of the next 30 years or so. Meanwhile, he spends time at his popular website Interactive Mathematics Miscellany and Puzzles.

BLL — The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

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