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

$a_n = (\sqrt{2})^n (c_1 \cos(\frac{3\pi}{4}n) + c_2 \sin(\frac{3\pi}{4}n))$

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

$a_n = (\sqrt{2})^n (c_1 \cos(\frac{\pi}{4}n) + c_2 \sin(\frac{\pi}{4}n))$