Count valid colourings on an hexagonal grid
Just a note (after the bounty expired) for the sake of future visitors:
Papers like "Bounds on the capacity of constrained two-dimensional codes" ( Forchhammer & Justesen, IEEE Transactions on Information Theory, 2000) and On the Capacity of 2-D Constrained Codes and Consequences for Full-Surface Data Channels (W. Weeks, 2012) and references cited therein could be a starting point. We could regard our hexagonal grid as a square grid and add run-length restrictions on diagonals.
Nevertheles, it seems hopeless to look for closed form solutions, but bounds and algorithms for estimating the capacity would be nice.