  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 $$ x_{n+1}
=(x_n^r +1)/{x_{n-1}} $$ 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.
