Solvability and solution character of a hyperbolic cotangent-type difference equation of second-order

In the present study, we show that the second-order nonlinear difference equation \(\begin{aligned} x_{n}=\frac{x_{n-1}x_{n-2}-a}{x_{n-1}-x_{n-2}},\ n\in \mathbb {N}_{0}, \end{aligned}\) $$\begin{array}{c}\hfill x_n=\frac{x_{n-1}{x}_{n-2}-a}{x_{n-1}-x_{n-2}},n\in {\mathbb{N}}_0,\end{array}$$ where \(a\in \left[ 0,\infty \right) \) $a\in \left[0,,,\infty \right)$ and the initial values \(x_{-2}\) $x_{-2}$ , \(x_{-1}\) $x_{-1}$ are real numbers, is solvable in closed form and that solutions can be analyzed in detail by the Fibonacci numbers.