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

$a_n = 1^n (A \cos(n) + B \sin(n))$

$a_n = A(1+i)^n + B(1-i)^n$

$a_n = (\sqrt{2})^n (A + Bn)$