<s>[INST] Recurrent equation system How to solve this recurrent equation system?
$$\begin{cases}a_{n+1}=a_n+2b_n\\
b_{n+1}=2a_n+b_n\end{cases}$$
The possible solutions should be
\begin{cases}a_n=\frac{1}{2}3^n+\frac{1}{2}(-1)^n\\
b_n=\frac{1}{2}3^n-\frac{1}{2}(-1)^n\end{cases}
 [/INST] $a_{n+1}=a_n+2(2a_{n-1}+b_{n-1})=a_n+4a_{n-1}+(a_n-a_{n-1})$
So, $a_{n+1}-2a_n-3a_{n-1}=0$.
The roots of $x^2-2x-3=0$ are $-1$ and $3$.
$a_n=\alpha (-1)^n+\beta(3)^n$ for some $\alpha$ and $\beta$.
$2b_n=a_{n+1}-a_n=\alpha(-1)^{n+1}+\beta(3)^{n+1}-\alpha (-1)^n-\beta(3)^n=-2\alpha(-1)^n+2\beta (3)^n$
$b_n=-\alpha(-1)^n+\beta (3)^n$
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