Recurrence relation $a_{n+1}=a_n^2-2$

Question

Sequence $a_n$ is defined $a_{n+1}=a_n^2-2, a_0=\alpha$.

I know that a closed form for $a_n$ is $a_n=\beta^{2^n}+\frac{1}{\beta^{2^n}}$, where $\beta$ satisfies $\beta+\frac{1}{\beta}=\alpha$ and I can prove it by induction.

But this way, we have to know answer. Is there any other solution that leads us to this form?

Answer 1

So far as I know, there is no general method for solving non-linear difference equations, even quadratic ones like this. The equation may suggest the clever transformation $$ \begin{align} a_n&=b_n+\frac1{b_n}\\ a_n^2&=b_n^2+\frac1{b_n^2}+2\\ a_{n+1}&=b_n^2+\frac1{b_n^2}\\ b_{n+1}&=b_n^2 \end{align}$$ but I don't know that I'd have come up with that if I hadn't seen the answer.

Answer 2

Since this is very close to a rival relation $a_{n+1}=a_n^2$ with easy solution $a_n=\alpha^{2^n}$, it's natural to try $a_n=\beta^{2^n}+\epsilon_n$ for some $\beta$ and sequence $\epsilon_n$ of small error terms. So$$\beta^{2^{n+1}}+\epsilon_{n+1}=(\beta^{2^n}+\epsilon_n)^2-2\implies\epsilon_{n+1}=2\beta^{2^n}\epsilon_n+\epsilon_n^2-2.$$This would be easy to solve if we only kept one of the last three terms, so let's try to cancel two of them. There are only so many combinations to try. You quickly realize cancelling the first and third gives the consistent conditions $\epsilon_n=\beta^{-2^n}$ and $\epsilon_{n+1}=\epsilon_n^2$. Then$$a_n=\beta^{2^n}+\beta^{-2^n}\implies\alpha=\beta+\beta^{-1}.$$

Answer 3

Hint: Use the double angle formula for cosine to show that if $a_n=2\cos\theta$, then $a_{n+1}=2\cos(2\theta)$.