A handbook of integer sequences.

*(English)*Zbl 0286.10001
New York-London: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers. xiii, 206 p. $ 10.00 (1973).

The late Leo Moser discussed the desirability of having a lexicographic list of integer sequences, so when I learned that the author had initiated such a project, I encouraged him to continue. The book is mainly a list of 2372 sequences. Each entry consists of a serial number; terms of the sequence; a name, descriptive phrase or recurrence and references to lead the user to a plethora of individual detail. Enough terms are given to occupy two lines of print, where they are known: the many cases where they are not represent challenges of unsolved problems: involving crystal lattices, map-folding, sorting, postage stamps, queens, no-3-in-line and crossing numbers; Ramsey numbers, Mersenne primes and least primitive roots; numbers of Boolean functions, graphs, groups, groupoids, knots, Latin squares, linear spaces, polyominoes, polytopes, groups, semigroups, squared rectangles, Steiner systems, threshold functions, topologies and walks.

To be included a sequence must consist of non-negative integers, be infinite (the Mersenne primes have been given the benefit of the doubt), start \(1,n,\ldots\) \((2\le n\le 999)\) (sequences have been doctored to achieve this, which may offend the purist, but some arbitrary decisions have to be made to ensure conciseness and unambiguity), be distinctive (e.g. \(1, 2, 4, 8, 16\) does not suffice; three sequences continue with 31, three with 32 and three with other terms) and pass the more subjective test of being well-defined and interesting.

Chapter II gives good advice for finding a sequence, by look-up, by modification, by recurrences, by differences and by factorization.

Chapter III depicts some of the more visual sequences: graphs, digraphs, trees, functional digraphs, topologies, posets, geometries, figurate numbers, slicing pancakes, necklaces, knots, stamp-folding and polyominoes and discusses Pólya enumeration, partitions, permutations, puzzle sequences, sequences from number theory, the numbers of Bell, Bernoulli, Euler and Stirling and the ubiquitous Catalan numbers. These last are the epitome of the unity of combinatorics, the best of numerous examples linking the various branches of a subject which appears to many to be no more than Poincaré’s “tas de pierres” instead of the noble mansion that it is.

This most useful collection has at once provided an economy of effort in research, a comparison of a large number of sequences revealing relations between seemingly disparate problems only slightly less spectacular than those exemplified by the Catalan numbers, and insights enabling renewed attacks on the many unsolved problems mentioned above. The index is well-prepared and gives another useful means of entry to the vast amount of information that is condensed within barely 200 pages.

The author (Math. Res. Center, Bell Telephone Labs, Murray Hill, N. J., U.S.A., 07974) is already issuing supplements to interested persons, containing new sequences and corrections and extensions to existing ones.

There are a few misprints and a few more serious errors. Sequences 330 and 1623 were miscalculated by their original authors; in fact 1622 and 1623 are the same sequence, another of the many examples of unification mentioned above. It is the problem solver’s dream and should provide enjoyment and possibly enlightenment even to the real mathematician.

{Editorial comments: Digitized expanded versions have appeared thereafter [Zbl 0845.11001] , especially valuable is the online version see https://oeis.org for details.}

To be included a sequence must consist of non-negative integers, be infinite (the Mersenne primes have been given the benefit of the doubt), start \(1,n,\ldots\) \((2\le n\le 999)\) (sequences have been doctored to achieve this, which may offend the purist, but some arbitrary decisions have to be made to ensure conciseness and unambiguity), be distinctive (e.g. \(1, 2, 4, 8, 16\) does not suffice; three sequences continue with 31, three with 32 and three with other terms) and pass the more subjective test of being well-defined and interesting.

Chapter II gives good advice for finding a sequence, by look-up, by modification, by recurrences, by differences and by factorization.

Chapter III depicts some of the more visual sequences: graphs, digraphs, trees, functional digraphs, topologies, posets, geometries, figurate numbers, slicing pancakes, necklaces, knots, stamp-folding and polyominoes and discusses Pólya enumeration, partitions, permutations, puzzle sequences, sequences from number theory, the numbers of Bell, Bernoulli, Euler and Stirling and the ubiquitous Catalan numbers. These last are the epitome of the unity of combinatorics, the best of numerous examples linking the various branches of a subject which appears to many to be no more than Poincaré’s “tas de pierres” instead of the noble mansion that it is.

This most useful collection has at once provided an economy of effort in research, a comparison of a large number of sequences revealing relations between seemingly disparate problems only slightly less spectacular than those exemplified by the Catalan numbers, and insights enabling renewed attacks on the many unsolved problems mentioned above. The index is well-prepared and gives another useful means of entry to the vast amount of information that is condensed within barely 200 pages.

The author (Math. Res. Center, Bell Telephone Labs, Murray Hill, N. J., U.S.A., 07974) is already issuing supplements to interested persons, containing new sequences and corrections and extensions to existing ones.

There are a few misprints and a few more serious errors. Sequences 330 and 1623 were miscalculated by their original authors; in fact 1622 and 1623 are the same sequence, another of the many examples of unification mentioned above. It is the problem solver’s dream and should provide enjoyment and possibly enlightenment even to the real mathematician.

{Editorial comments: Digitized expanded versions have appeared thereafter [Zbl 0845.11001] , especially valuable is the online version see https://oeis.org for details.}

Reviewer: Richard K. Guy

##### MSC:

11-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to number theory |

05-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to combinatorics |

11Bxx | Sequences and sets |

05Axx | Enumerative combinatorics |

11P81 | Elementary theory of partitions |

65A05 | Tables in numerical analysis |