On the density of certain languages with $p^2$ letters

The sequence $(x_n)_{ninmathbb N} = (2,5,15,51,187,dots)$ 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 $(mathbb Z_2times mathbb Z_2)^n$ 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 $(mathbb Z_ptimesmathbb Z_p)^n$ and a language with $p^2$ letters and words of length $n$.