On the density of certain languages with \(p^2\) letters (Q2517662)

Summary: The sequence \((x_n)_{n\in\mathbb N} = (2,5,15,51,187,\dots)\) given by the rule \(x_n=(2^n+1)(2^{n-1}+1)/3\) appears in several seemingly unrelated areas of mathematics.~For example, \(x_n\) is the density of a language of words of length \(n\) with four different letters.~It is also the cardinality of the quotient of \((\mathbb Z_2\times \mathbb Z_2)^n\) under the left action of the special linear group \(\mathrm{SL}(2,\mathbb Z)\).~In this paper we show how these two interpretations of \(x_n\) are related to each other.~More generally, for prime numbers \(p\) we show a correspondence between a quotient of \((\mathbb Z_p\times\mathbb Z_p)^n\) and a language with \(p^2\) letters and words of length \(n\).