The global attractivity of the rational difference equation $y_{n}=1+\frac{y_{n-k}}{y_{n-m}}$

This paper studies the behavior of positive solutions of the recursive equation

$y_{n}=1+\frac{y_{n-k}}{y_{n-m}}, ~~ n=0,1,2,\ldots,$

with $y_{-s},y_{-s+1}, \ldots, y_{-1} \in (0, \infty)$ and $k,m \in \{1,2,3,4,\ldots\}$, where $s=\max\{k,m\}$. We prove that if $\mathrm{gcd}(k,m) = 1$, with $k$ odd, then $y_n$ tends to $2$, exponentially. When combined with a recent result of E. A. Grove and G. Ladas (Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC Press, Boca Raton (2004)), this answers the question when $y=2$ is a global attractor.