On the nature of solutions of the difference equation $\mathbf{x_{n+1}=x_{n}x_{n-3}-1}$


1 Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, P. O. Box 842014, Richmond, Virginia 23284-2014 USA.

2 Department of Mathematical Sciences, Appalachian State University, Boone, North Carolina 28608 USA.


We investigate the long-term behavior of solutions of the difference equation
$$x_{n+1}=x_{n}x_{n-3}-1,   n=0,1, \ldots $$
where the initial conditions $x_{-3} ,, x_{-2} ,, x_{-1} ,, x_{0}$ are real numbers.  In particular, we look at the periodicity and asymptotic periodicity of solutions, as well as the existence of unbounded solutions.