Some people collect stamps, other coins, phone cards, beer mats or butterflies. There is hardly anything which has not become the object of the human passion to collect, even chamber pots and shoe laces have found their devotees. But the American mathematician Neil J. A. Sloane of the AT&T Shannon Labs in Florham Park/New Jersey has probably chosen the most unusual objects to collect. He collects integer sequences. Not any arbitrary sequences however, but only those which consist of positive integers, which have infinitely many elements, and which are formed according to a fixed rule.

Although Sloane is probably the only collector of integer sequences in the world, his hobby has met with wide interest. Thousands of scientists and amateurs have been helping him for many years now to continuously extend his collection. In December 1963 Sloane, who was at this time still a student at Cornell University in Ithaca/New York, looked for information about a certain sequence from graph theory. But as hard as he tried, he could not find anything about it in the relevant literature. That annoyed him so much that he began to collect sequences systematically.

Ten years later his collection contains over 2300 sequences from all areas of mathematics, the natural sciences and even from puzzles. He arranged them lexicographically and published them as book with the title "A Handbook of Integer Sequences." The book became a success, and many people sent him new sequences. Neil Sloane continued to collect. Together with Simon Plouffe of the Université du Québec in Montréal in 1995 he wrote the "Encyclopedia of Integer Sequences", which with 5488 sequences was more than twice as large as his first collection.

In the same year Sloane created e-mail addresses with which one could make automatic look-ups in his sequence database. The book and the e-mail addresses were a great success and led to an enormous wave of contributions of new sequences. One year later the collection had already increased to 16,000 sequences. Then Sloane also created an Internet page for his sequence database with special search functions (http:oeis.org). The interest among scientists and also among amateurs is enormous. Every day his collection is accessed about 2,500 times, which now contains over 61,000 sequences.

Sloane's collection resembles a well-stocked department store. All conceivable sequences are to be found there. Mathematical sequences such as the prime numbers (2, 3, 5, 7, 11 ...), the square numbers (0, 1, 4, 9, 16 ...) or the factorials (1, 1, 2, 6, 24 ...) are of course copiously represented. In addition, Neil Sloane has included sequences from chemistry like the number of different alkanes with n carbon atoms (1, 1, 1, 2, 3, 5 ...), or sequences from physics like the number of Feynman diagrams of order 2n (1, 3, 18, 153, 1638...), as well as sequences from biology like the possible secondary structures of a RNA molecule with n nucleotides (1, 1, 1, 2, 4, 8, 17 ...).

Additionally, the collection contains chess problems like the number of ways of placing n queens on a chessboard with n by n squares in such a way that they do not attack each other (1, 0, 0, 2, 10, 4, 40, 92 ...). One can also find curiosities like the sequence 0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55 ... It results from removing all letters except the number characters I, V, X, L, C, D and M from the English numbers one, two, three, four, five, ... The resulting words are then interpreted as Roman numbers.

The question "what is the next number?" in a given sequence, which is popular in puzzle columns and intelligence tests, is easy to solve with Sloane's collection. For example, if one enters the sequence 3, 1, 4, 1, 5 into the search program, it offers thirty-five different ways to continue the sequence. One of the resulting sequences would be the one consisting of all natural numbers, beginning with three, separated by ones. Thus the next number would have to be one. But it could also represent the decimals of the number pi. Then the next element would have to be nine. Since 1998 the American mathematician even publishes a special electronic journal, the "Journal of Integer Sequences", which contains exclusively articles on integer sequences.

HEINRICH HEMME

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Translated from the German by Reiner Martin.