$a_n = c_1(1+2i)^n + c_2(1-2i)^n$

$a_n = r^n(c_1 \cos(n\theta) + c_2 \sin(n\theta))$ where $r = \sqrt{5}$

$a_n = c_1 \cos(n\theta) + c_2 \sin(n\theta)$

$a_n = n(c_1 \cos(n\theta) + c_2 \sin(n\theta))$