Application of logic to integer sequences: A survey

Abstract

Chomsky and Schützenberger showed in 1963 that the sequence d L(n), which counts the number of words of a given length n in a regular language L, satisfies a linear recurrence relation with constant coefficients for n, or equivalently, the generating function gL(x) = ΣndL(n)x3 is a rational function. In this talk we survey results concerning sequences a(n) of natural numbers which - satisfy linear recurrence relations over ℤ or ℤm, and - have a combinatorial or logical interpretation. We present the pioneering, but little known, work by C. Blatter and E. Specker from 1981, and its further developments, including results by I. Gessel (1984), E. Fischer (2003), and recent results by T. Kotek and the author. © 2010 Springer-Verlag.