A word has a square if it contains two consecutive occurrences of the same factor. For an alphabet of size k ≥ 3, we ask for the limit α = lim_{n → ∞} (log S(n))/n, where S(n) denotes the number of squarefree words of length n. The problem is most interesting in the case of k = 3, i.e., of three letters. In this case it is known that 1.3017597 < α < 1.3017619. See, for example, the review of Shur in Computer Science Review 6 (2012) 187-208.

-- JeffreyShallit - 13 Jul 2011