  Snake graphs are a class of planar graphs that are important in the theory of
cluster algebras. Indeed, the Laurent expansions of the cluster variables in
cluster algebras from surfaces are given as weight generating functions for
1-dimer covers (or perfect matchings) of snake graphs. Moreover, the
enumeration of 1-dimer covers of snake graphs provides a combinatorial
interpretation of continued fractions. In particular, the number of 1-dimer
covers of the snake graph $\mathscr{G}[a_1,\dots,a_n]$ is the numerator of the
continued fraction $[a_1,\dots,a_n]$. This number is equal to the top left
entry of the matrix product $\left(\begin{smallmatrix} a_1&1\\1&0
\end{smallmatrix}\right) \cdots \left(\begin{smallmatrix} a_n&1\\1&0
\end{smallmatrix}\right)$.
  In this paper, we give enumerative results on $m$-dimer covers of snake
graphs. We show that the number of $m$-dimer covers of the snake graph
$\mathscr{G}[a_1,\ldots,a_n]$ is the top left entry of a product of analogous
$(m+1)$-by-$(m+1)$ matrices. We discuss how our enumerative results are related
to other known combinatorial formulas, and we suggest a generalization of
continued fractions based on our methods. These generalized continued fractions
provide some interesting open questions and a possibly novel approach towards
Hermite's problem for cubic irrationals.
