Long-Term Behavior of Solutions of the Difference Equation x n + 1 = x n − 1 x n − 2 − 1

Abstract

We investigate the long-term behavior of solutions of the following difference equation: $x_{n+1}=x_{n-1}{x}_{n-2}-1$, $n\in {\mathbb{N}}_0$, where the initial values $x_{-2}$, $x_{-1}$, and $x_0$ are real numbers. Numerous fascinating properties of the solutions of the equation are presented.