The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type $A_n$, by counting the mutation class of any quiver with underlying graph $A_n$. It will also follow that if $T$ and $T'$ are cluster-tilting objects in a cluster category $\mathcal{ C}$, then $\End_{ \mathcal{ C}}(T)$ is isomorphic to $\End_{ \mathcal{ C}}(T')$ if and only if $T=\tau^i T'$.
Categories: math.RT