Let \{g_i\}_{i=0}^{\infty} be a sequence of positive integers such that g_0 = g_1 = 1 and the following recursions hold for every positive integer n:
$$g_{2n+1} = g_{2n-1}^2 + g_{2n-2}^2$$
$$g_{2n} = 2g_{2n-1}g_{2n-2} - g_{2n-2}^2$$
Compute the remainder when g_{2011} is divided by 216.