  We consider the Ptolemy cluster algebras, which are cluster algebras of
finite type $A$ (with non-trivial coefficients) that have been described by
Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed
$\zS$ in a Ptolemy cluster algebra, we present a formula for the expansion of
an arbitrary cluster variable in terms of the cluster variables of the seed
$\zS$. Our formula is given in a combinatorial way, using paths on a
triangulation of the polygon that corresponds to the seed $\zS$.
