I'm looking at the sequence
$x_{n+1}=x_n-1/x_n$
and want to find starting values $x_0$ for which the sequence is periodic. I want to calculate all values for which the sequence is periodic with a period $p<P$.
The problem can be rewritten as:
$f(x)=x-1/x$
and $f^p$ is the function applied $p$-times.
To calculate the values for which the sequence is periodic I tried to find the $x_0$ for which $f^{p}(x_0)=x_0$ by looking at the roots of $f^{p}(x_0)-x_0$. Since the degree of the polynomial is ~$2^p$ (look Related question) this approach is not viable for $p>10$.
More Background info:
I'm interested in finding the range of the periodic sequences so for a given cycle
$S={x_0,...,x_n}$
where $f(x_n)=x_0$
I want to calculate the range:
$r=max(S)-min(S)$
So any ideas or hints on how to calculate the range (directly) or how to get the values for which the sequence is periodic will be greatly appreciated.