An Identity for Pell-numbers The Pell-numbers are defined recursively by:
$P_0 = 0,  
P_1 = 1$ and $
P_{n+2} = 2P_{n+1} + P_n$
I am stuck trying to prove the identity:
$P_{2n+1}^2 - 1 = 4P_n P_{n+1}(P_{n+1}^2 - P_n^2)$ 
A proof would be great, otherwise a list of known Pell-identies would also help

You know $P_{n+2} = 2 P_{n+1} + P_{n}$ $\Rightarrow$ $P_{n+2}-(1+\sqrt{2})P_{n+1}= (1-\sqrt{2})(P_{n+1}-(1+\sqrt{2})P_{n})$. In other word, we have 
\begin{align*}
\frac{P_{n+2}-(1+\sqrt{2})P_{n+1}}{P_{n+1}-(1+\sqrt{2})P_{n}} = (1-\sqrt{2}).
\end{align*}
Since $P_0 = 0, P_1 = 2, P_2 = 2$, we deduce that 
\begin{align*}
P_{n} = \frac{(1+\sqrt{2})^n - (1-\sqrt{2})^n }{2\sqrt{2}}.
\end{align*}
You refer to Pell_number.
