Question 93
Let be given the sequence $\left(x_{n}\right)$ defined by $x_{1}=1$ and $x_{n+1}=3 x_{n}+\left\lfloor x_{n} \sqrt{5}\right\rfloor$ for all $n=1,2,3, \ldots$, where $\lfloor x\rfloor$ denotes the greatest integer that does not exceed $x$. Prove that for any positive integer $n$ we have
$$
x_{n} x_{n+2}-x_{n+1}^{2}=4^{n-1}
$$
$$
x_{n} x_{n+2}-x_{n+1}^{2}=4^{n-1}
$$