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SIAM Review of ``An Encyclopedia of Integer Sequences'' by N. J. A. Sloane & Simon Plouffe

J. M. Borwein and Robert M. Corless

In SIAM REVIEW, 38, (1996), 333-337.

``The Encyclopedia of Integer Sequences'' by Sloane and Plouffe, published by Academic Press, is not a normal book. It contains a lexicographically ordered collection of integer sequences together with references to these sequences where they appear in the literature. The idea is that a researcher who encounters a sequence in her or his work, and wishes to quickly find out what is known about the sequence (does it have a name, for example, such as ``the Euler numbers'' or ``the Stirling numbers of the first kind''?), can look it up here.

On the face of it this seems a difficult task to accomplish, because surely there are very many sequences of interest. However, by Pareto's principle (80% of your work is done with 20% of your tools) we would expect that simple sequences would occur often, and thus such a book would be useful.

Indeed, this is the case, and even if the book were no more than the handsomely bound physical collection it is, it would have been worthwhile to create, publish, or buy, because it provides a very cheap and efficient route to answers that will work sometimes: if it doesn't work on a particular problem, no big effort has been expended, while if it does work you may save a lot of time.

But the physical book is not the whole story. Sloane and Plouffe have also created two ``avatars'' of the book, as freely available online computer programs (which we will call sequences and superseeker) for people to send their sequences to. Because the programs can be accessed by people who do not own the book, we think that Academic Press deserves considerable praise for its enlightened attitude towards the changing shape of publishing.

This is not the first, but is one of the first, of a growing list of sophisticated tools which are accessible to even relatively naive users, and which dramatically illustrate a positive use of the Internet. Our dream work environment would provide us with a whole palette of such tools and a simple key to what exists and how to use it. These tools should ideally be immediately integratable with your favourite working environment (MATLAB, Axiom, Maple, Mathematica, Whatever).

In our opinion the physical book is itself worthwhile not only because it is pleasant to browse in (electrons are so cold, in comparison) but also because of the discussion at the beginning on analysis of sequences. Some of the heuristics discussed in chapters 1, 2, and 3 (before the table of sequences proper begins) give useful hints as to what to do when the computer programs don't work; they also give a nice conceptual model of the inner workings of the programs.

One can turn the tables (so to speak) and use the sequences from the book as a test of each of the subprograms in sequences and superseeker. Simon Plouffe tells us that each subprogram was considered useful enough to be included if it could identify on the order of 10-100 of the sequences from the book. Further, about 25% of the sequences in the book are obtained from a rational generating function or elementary manipulation thereof (reversion, undoing a logarithmic differentiation, etc.). Addition of various other classes such as hypergeometric functions and pre-processing (adding `1' to each term or doubling the terms, etc) significantly increased the hit rate. It is to be emphasized that not every plausible transformation was included, and much expertise on the part of the authors was needed to choose useful transformations and to avoid `the curse of exponentiality'.

Finally, some `off-the-wall' sequences are also included, such as the numbers on the New York Subway stops, in Figure M5405.

Incidentally, due to a printer's error the table of Figures was not included in the book, and as the `silly' sequences are not actually indexed or numbered in the book, one must either use the programs or know that they are contained in Figure M5405 to find them.

We now give some examples of the uses of the book and the programs therein, to demonstrate their utility (and also some limitations).

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