Limit by diagonalisation I want to compute the limit of the sequence $x_0=a$, $x_1=b$ and $x_n=\frac{x_{n-1}+x_{n-2}}{2}$.
I did it by setting:
$\begin{pmatrix}
x_{n+1}\\ x_{n}
\end{pmatrix}=$
$\begin{pmatrix}
\frac{1}{2} &\frac{1}{2}\\
1 &0\\
\end{pmatrix}$
$\begin{pmatrix}
x_{n}\\ x_{n-1}
\end{pmatrix}$ 
and calculating, that $$\lim\limits_{n\to\infty}\begin{pmatrix}
\frac{1}{2} &\frac{1}{2}\\
1 &0\\
\end{pmatrix}^n=\begin{pmatrix}
\frac{2}{3} &\frac{1}{3}\\
\frac{2}{3} &\frac{1}{3}\\
\end{pmatrix}$$
And thus $\lim\limits_{n\to\infty}x_n=\frac{2}{3}b+\frac{1}{3}a$.
But this can't be true, because the result should be symmetric in $a$ and $b$. Where is my mistake?

$x_2 = \dfrac{a + b}{2}$
$x_3 = \dfrac{a+3b}{4}$
It is already not symmetric in $a$ and $b$. There is no error.
