A Combinatorial Formula for Rank 2 Cluster Variables

Let $r$ be any positive integer, and let $x_1,x_2$ be indeterminates. We consider the sequence $\{x_n\}$ defined by the recursive relation $$x_{n+1}=(x_n^r+1)\mathrm{/}{x}_{n-1}$$ for any integer $n$. Finding a combinatorial expression for $x_n$ as a rational function of $x_1$ and $x_2$ has been an open problem since 2001. We give a direct elementary formula for $x_n$ in terms of subpaths of a specific lattice path in the plane. The formula is manifestly positive, providing a new proof of a result by Nakajima and Qin.