$a_n = \frac{1}{3} 2^n + \frac{2}{3} (-1)^n$

$a_n = \frac{2}{3} 2^n + \frac{1}{3} (-1)^n$

$a_n = 2^n + (-1)^n$

$a_n = \frac{1}{2} 2^n + \frac{1}{2} (-1)^n$