Some sequences of integers

What I want to describe is a kind of experimental mathematics, ideal for doing at times when honest thinking is not going well. The requirements are a small computer (pencil and paper suffice, though the calculations are tedious), and Neil Sloane’s “A Handbook of Integer Sequences” [15]. This book, a kind of hitch-hikers’ guide to the universe N”, consists mainly of a list of 2372 sequences of nonnegative integers, arranged lexicographically, with an index, references, and notes for users. The main criterion for inclusion of a sequence is that somebody must have found it sufficiently interesting to record it in the literature. The Handbook can be used, then, like a book of tables, using the index to locate a sequence. A more exciting possibility is this. Suppose you find youself in possession of an “unknown” sequence. (This is not an uncommon event; a glance through the Handbook confirms that sequences occur in all provinces of mathematics, and well beyond its frontiers.) If you can locate your sequence in the Handbook, you have both a problem (of showing that your sequence really is the one listed) and a source of information (the references to the sequence). I know of several cases where new results have been discovered this way. I propose a third way of using the Handbook. There are some naturallyoccurring transformations of sequences, two of which I will consider in detail. Finding instances where a known sequence is transformed into another can give rise to new mathematical insights in the way described above. Also any sequence which is transfirmed into a closely-related one gains significance independent of the objects it counts. Sloane adops the convection that all sequences commence 1, IZ, when n > 1. To ensure this, he deletes “superfluous” leading ones and zeros, and inserts a 1 if necessary. Some valuable information is lost in this way, namely the “natural” starting point of the sequence. But, on the positive side, the weakness of the