##
**Concrete mathematics. A foundation for computer science.**
*(English)*
Zbl 0668.00003

Reading, MA: Addison-Wesley Publishing Company. xiii, 625 p. (1989).

In the continuing debate on what should constitute an undergraduate background in mathematics for the mathematician or the computer scientist, this book is destined for a major role. It is not a traditional course in discrete mathematics, though it does overlap some of the material taught there. Rather it is a compendium of some of the basic techniques for solving the very specific problems that arise in all branches of the mathematical sciences. The heart of the book presents elementary number theory, binomial coefficient identities, topics from combinatorics and probability, and a primer on how to use and find asymptotic arguments.

The style is extremely informal, often irreverent. But it also conveys the joy and exuberance of the craftsman who is only too pleased to share his lore by shaping a well-turned example before your eyes.

My only reservation concerns the chapter on binomial coefficient identities, by far the longest in the book. While the authors do introduce hypergeometric functions, a standardized notation for binomial coefficient identities that quickly enables you to identify when the summation you are pondering is equivalent to one that has been previously studied, they wait until they are more than fifty pages into the subject before letting the reader in on the existence of this notation. They could have brought it in earlier.

The style is extremely informal, often irreverent. But it also conveys the joy and exuberance of the craftsman who is only too pleased to share his lore by shaping a well-turned example before your eyes.

My only reservation concerns the chapter on binomial coefficient identities, by far the longest in the book. While the authors do introduce hypergeometric functions, a standardized notation for binomial coefficient identities that quickly enables you to identify when the summation you are pondering is equivalent to one that has been previously studied, they wait until they are more than fifty pages into the subject before letting the reader in on the existence of this notation. They could have brought it in earlier.

Reviewer: David M. Bressoud (Saint Paul)

### MSC:

00A05 | Mathematics in general |

68-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science |

11B65 | Binomial coefficients; factorials; \(q\)-identities |

11Axx | Elementary number theory |

05A10 | Factorials, binomial coefficients, combinatorial functions |

### Keywords:

basic techniques of mathematical sciences; elementary number theory; binomial coefficient identities; combinatorics; probability; asymptotic arguments
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\textit{R. L. Graham} et al., Concrete mathematics. A foundation for computer science. Reading, MA: Addison-Wesley Publishing Company (1989; Zbl 0668.00003)

### Online Encyclopedia of Integer Sequences:

Least k such that H(k) > n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.Number of fractions in Farey series of order n.

Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.

Continued fraction expansion of Fibonacci factorial constant.

Decimal expansion of Fibonacci factorial constant.

Increasing partial quotients of e^gamma = 1.78107241799019798523650...

Triangle read by rows giving coefficients in Bernoulli polynomials as defined in A001898, after multiplication by the common denominators A001898(n).

A variant of Josephus Problem in which 2 persons are to be eliminated at the same time.

Product of first n nonzero even-indexed Fibonacci numbers F(2), F(4), F(6), ..., F(2*n).

Product of first n nonzero odd-indexed Fibonacci numbers F(1), ..., F(2*n-1).

Constant associated with the product of the first n nonzero even-indexed Fibonacci numbers.

Constant associated with the product of the first n nonzero odd-indexed Fibonacci numbers.