A maxmin problem on finite automata (Q1116710)

We solve the following problem proposed by Straubing. Given a two-letter alphabet A, what is the maximal number of states f(n) of the minimal automaton of a subset of \(A^ n\), the set of all words of length n. We give an explicit formula to compute f(n) and we show that \(1=\liminf_{n\to \infty}nf(n)/2^ n\leq \limsup_{n\to \infty}nf(n)/2^ n=2.\)