Let $X_0, X_1$ and $\{a_n,n\geq0\}$ are i.i.d. $\sim $Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1 $$ Then it follows that $\{\mathbb{E}[X_n]\}$ is a Fibonacci sequence. But I could infer nothing for either $\mathbb{E}[X_{n}^k]$ for $k>1$ or for the characteristic function of $X_n$, $n>1$. Any idea?
Edit: I forgot to mention my progress. Here is what I could do:
$$\mathbb{E}(e^{tX_{n+1}})=\mathbb{E}(e^{ta_{n}X_n})\mathbb{E}(e^{ta_{n-1}X_{n-1}})$$ Now, $$\mathbb{E}(e^{ta_{n}X_n})=\frac{1}{2}\left(\mathbb{E}(e^{2tX_n})+1\right)$$ Denoting $\mathbb{E}(e^{tX_n})=:m_n(t)$, we get the recurrence, $$m_{n+1}(t)=\frac{1}{4}(1+m_n(2t))(1+m_{n-1}(2t))$$