It is not 256.
@nbouscal's way is closest to my original method, although I would be very interested in seeing if
@asnaseer's way also pans out.
I will point out that the solution is an odd number, however odd that sounds (pun intended). In asnaseer's potential solution, after you multiply by 2, you are counting 8 purple/8 green twice, so you have to subtract it (I think that should work).
Hint containing my method in white \(\LaTeX\) below: \[\color{white}{\text{I used Burnside's Lemma from Group Theory using the group } D_8.}\]\[\color{white}{D_8\text{ is also called }D_{16}\text{ sometimes.}}\]