how many unique patterns can be made on a 4x4 grid accounting for rotational symmetry?

Question

I was working with higher range Margolus cellular automata neighborhoods and i ran into a problem, how many patterns you can make in a 4x4 square grid if you consider rotationally symmetric patterns to be the same, but mirror symmetric patterns to be separate? additionally, if someone could provide an equation to calculate how many there are in an arbitrary NxN neighborhood, that would be greatly appreciated. i know that the total number of anisotropic patterns is 65536, but beyond that i am unsure.

Answer 1

Suppose you have $k$ states and an $n×n$ grid. There are four automorphisms: the identity (fixes $k^{n^2}$ grids), a $180^\circ$ rotation (fixes $k^{\lceil n^2/2\rceil}$ grids and two $90^\circ$ rotations (fix $k^{\lceil n^2/4\rceil}$ grids each). Thus, by Burnside's lemma, the answer is $$\frac14(k^{n^2}+k^{\lceil n^2/2\rceil}+2k^{\lceil n^2/4\rceil})$$ For $k=2$ and $n=4$ this works out as $\frac14(2^{16}+2^8+2\cdot2^4)=16456$.