Consider the sequence defined by $a_n = 2a_{n-1} - 2a_{n-2}$ with $a_0 = 0$ and $a_1 = 2$. The characteristic equation $r^2 - 2r + 2 = 0$ has roots $r = 1 \pm i$, which in polar form are $r = \sqrt{2}e^{\pm i\pi/4}$. Using Euler's formula, compute $a_5$.