On the nature of solutions of the difference equation \mathbf{x_{n+۱}=x_{n}x_{n-۳}-۱}

We investigate the long-term behavior of solutions of the difference equationx_{n+۱}=x_{n}x_{n-۳}-۱,   n=۰,۱, \ldots where the initial conditions x_{-۳} ,, x_{-۲} ,, x_{-۱} ,, x_{۰} are real numbers.  In particular, we look at the periodicity and asymptotic periodicity of solutions, as well as the existence of unbounded solutions.

Keywords:

Difference equations , boundedness , periodicity , Asymptotic periodicity , Eventual periodicity , Invariant interval , Continued fractions