I have the sequence defined by,
$a_{n+2} = \frac{1}{3} a_{n+1} + \frac{2}{3}a_n$,
where $a_0 = 0, a_1 = 1$.
I have proved that this converges, but I cannot figure out what it converges to. I tried to find some pattern in the terms so that I could find a closed-form of each term, but I had no luck.