For $\alpha \in \mathbb{R}$, define the sequence $\{x_n \}$ by $x_1 = \alpha$, and $x_{n+1} = x_n^2 - 1 $. It is true that if the sequence converges, then it must converge to $ (1 \pm \sqrt{5} ) /2$. Find all values of $\alpha$ s.t. the sequence converges to $ (1 + \sqrt{5} ) /2$.
My main line of attack that I kept returning to was to try to express $x_n$ as a function of $\alpha$, so as to determine what properties $\alpha$ must have for the desired convergence. But the sequence becomes very unweilding as an explicit function of $\alpha$. I've also discovered that convergence will fail for values of $\alpha \in \{1, -1, 0, \}$, but also have trouble determining the all the bad values of $\alpha$. Hints appreciated.