Patterns and Fractions

By Aaron Robertson, Herbert Wilf, and Doron Zeilberger

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Written: April 17, 1999

(Appeared in Electronic J. Combinatorics v. 6 (1999), R38).

This article started out as a heavy-duty symbol-crunching sequel of a previous human-insight-heavy article by Aaron. Then Herb came along, and using beautiful human insight, found an EXPLICT and AMAZING expression, in terms of a Rogers-Ramanujan-like continued fraction, for the generating function, which, as a bonus, implied a super-fast algorithm for computing what we call in the paper AR(r) and Aaron(r). Just for comparison, with the original, slow, algorithm (that is kept in the Maple version, downloadable below, for old-time's sake, and for comparison and check, as AaronSlow), it took all night to find Aaron(4), and now it takes a few seconds. Just type `Aaron(4);' .

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