The encyclopedia of integer sequences. Incl. 1 IBM/MS-DOS disk. (3.5).

*(English)*Zbl 0845.11001
San Diego, CA: Academic Press. xiii, 587 p. (1995).

This is a greatly expanded version of the first author’s “A Handbook of Integer Sequences” [Academic Press, New York (1973; Zbl 0286.10001)]. The main table (pp. 37–534) is a list of 5488 lexicographically ordered sequences, giving, for each sequence, an identification number, two lines of sequence terms, a name or a descriptive phrase, a recurrence relation (if known), and some additional information including references. For 58 particularly interesting sequences, additional information is presented in so-called Figures. For example, there are Figures for Pascal’s triangle, Bernoulli numbers, graphs, polygonal numbers and RNA molecules. One purpose of these Figures is to display important arrays of numbers like Pascal’s triangle. In order to be included in the table a sequence must be “well-defined and interesting” in the sense that it should have appeared in the scientific literature.

This book not only is of great help in identifying known integer sequences, it also provides, in a separate chapter, a wealth of information and tips to study and identify sequences which are not in the table.

Two on-line versions of the Encyclopedia are available through electronic mail. The first is a simple look-up service (send email to sequences@research.att.com simply containing a line of the form lookup 5 14 42 132 429) and the second tries very hard to find an explanation for a given sequence (send email to superseeker@research.att.com with a line of the form lookup 1 2 4 6 10 14 20 26 36 46 60 74 94 114 140 166).

The reviewer has tested the second on-line version with the sequence defined by: \(u_0= 0\), \(u_1= 1\), \(u_{n+ 1}= u_n+ u_{n- 1}\) if \(n\) is even, and \(u_{n+ 1}= 3u_n -u_{n- 1}\) if \(n\) is odd [taken from: J. L. Simons, Conditional recurring sequences, Thesis, Delft University of Technolgy, Delft (1976)]. A few seconds after sending the message: lookup 1 3 4 9 13 30 43 99 142 327, an email answer arrived telling (a.o.) that this sequence does not seem to be in the table, but that it may have the generating function \((1+ 3x+ 3x^2)/(1- 3x^2- x^4)\). Indeed, this was easy to check, so we learned that the conditional recursion could be replaced by the unconditional recursion: \(u_{n+ 2}= 3u_n- u_{n- 2}\), for \(n\geq 2\).

It is unavoidable that an encyclopedic work like this contains errors. Fortunately, the first author gives actual error update information on his web page.

Editorial note: One should now use the online-version available at

http://oeis.org/

This book not only is of great help in identifying known integer sequences, it also provides, in a separate chapter, a wealth of information and tips to study and identify sequences which are not in the table.

Two on-line versions of the Encyclopedia are available through electronic mail. The first is a simple look-up service (send email to sequences@research.att.com simply containing a line of the form lookup 5 14 42 132 429) and the second tries very hard to find an explanation for a given sequence (send email to superseeker@research.att.com with a line of the form lookup 1 2 4 6 10 14 20 26 36 46 60 74 94 114 140 166).

The reviewer has tested the second on-line version with the sequence defined by: \(u_0= 0\), \(u_1= 1\), \(u_{n+ 1}= u_n+ u_{n- 1}\) if \(n\) is even, and \(u_{n+ 1}= 3u_n -u_{n- 1}\) if \(n\) is odd [taken from: J. L. Simons, Conditional recurring sequences, Thesis, Delft University of Technolgy, Delft (1976)]. A few seconds after sending the message: lookup 1 3 4 9 13 30 43 99 142 327, an email answer arrived telling (a.o.) that this sequence does not seem to be in the table, but that it may have the generating function \((1+ 3x+ 3x^2)/(1- 3x^2- x^4)\). Indeed, this was easy to check, so we learned that the conditional recursion could be replaced by the unconditional recursion: \(u_{n+ 2}= 3u_n- u_{n- 2}\), for \(n\geq 2\).

It is unavoidable that an encyclopedic work like this contains errors. Fortunately, the first author gives actual error update information on his web page.

Editorial note: One should now use the online-version available at

http://oeis.org/

Reviewer: H. J. J. te Riele (Amsterdam)

##### MSC:

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

11Y55 | Calculation of integer sequences |

11Bxx | Sequences and sets |

11B83 | Special sequences and polynomials |

11B37 | Recurrences |

11B68 | Bernoulli and Euler numbers and polynomials |