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

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

$c_1 \cos(n) + c_2 \sin(n)$

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