Thinking about the nature of the beast, I'd bet on powers
of 4 having extra solutions and the squares that that have
4xN+1 sides (5, 9, 13, 17) having extra solutions as well.
I think the 16x16 will have at least 2**16 solutions since
it is actually a 2x2 of 4x4s and the 4x4 had 2**4 solutions.
It also wouldn't suprise me that 13x13 had 8**8 solutions.