An Enumeration Problem Arising From a Result About Arithmetic Progressions

For positive integers m, n, q, g(m, n, q) is the number of words of length n over an alphabet A = {a(1)...,a(q)} such that there is no block of m consecutive a(q)'s in the word. A recent result shows that when q is a prime and m <= n then g(m, n, q) is a lower estimate of the cardinality of the largest set of integers in {0,...,q(n) - 1} which contains no q(m)-term arithmetic progression. We give formulas for and estimates of g(m, n, q) in special cases, and also a linear difference equation satisfied by g(m, n, q) as a function of n.