Consider a sequence ${a_n}$ satisfying the recurrence relation $$a_n= \frac{1}{P-a_{n-1}}, ~~~~ a_1=\frac{1}{P}$$ where $P\geq 2$ is an integer. Is there an analytic expression of the value of $a_{n\to \infty}$?
For example, we can find that for $P=2$, $a_n= \frac{n}{n+1}$, hence $a_{n\to \infty}=1$. For $P\geq 3$, mathematica gives $a_n$ approaches 0.381966 upto $n=1500$. I am not sure if there exists an analytical expression for $a_{n\to \infty}$ in terms of $P$. Any comments are welcome!
If the sequence were to converge to $x$, then one would have that $$x=\frac{1}{P-x}$$ thus $$x^2-Px+1=0$$ from which $$x=\frac{P\pm\sqrt{P^2-4}}{2}$$