Experimental methods applied to the computation of integer sequences

Abstract

We apply techniques of experimental mathematics to certain problems in number theory and combinatorics. The goal in each case is to understand certain integer sequences, where foremost we are interested in computing a sequence faster than by its definition. Often this means taking a sequence of integers that is defined recursively and rewriting it without recursion as much as possible. The benefits of doing this are twofold. From the view of computational complexity, one obtains an algorithm for computing the system that is faster than the original; from the mathematical view, one obtains new information about the structure of the system. Two particular topics are studied with the experimental method. The first is the recurrence an=an-1 +gcdn,an-1 , which is shown to generate primes in a certain sense. The second is the enumeration of binary trees avoiding a given pattern and extensions of this problem. In each of these problems, computing sequences quickly is intimately connected to understanding the structure of the objects and being able to prove theorems about them.