Explicit formula from for interesting reccurence $$t_0=-5, t_1=\frac{11}{5} $$
$$t_n=1-\frac{2}{t_{n-1}} + \frac{-4}{t_{n-2}*t_{n-1}} $$
I have never seen so uncommon formula for recurrence like this before. I have no idea how to solve it. Please, can anyone help me ?

The recurrence can be re-written as $t_n t_{n-1} t_{n-2} = t_{n-1}t_{n-2} - 2t_{n-2} -4$.
Let $x_n = t_0 t_1 t_2 \dotsb t_n$. Then
$$\frac{x_n}{x_{n-3}} = \frac{x_{n-1}}{x_{n-3}} - 2\frac{x_{n-2}}{x_{n-3}} -4$$
Hence $x_{n} = x_{n-1} - 2x_{n-2} - 4x_{n-3}$ and $x_0 = -5$, $x_1 = -11$, $x_2 = -5$. You can find $x_n$ from this (one solution of the characteristic equation is $r=-1$), and then $t_n = \frac{x_{n}}{x_{n-1}}$. 
