HMMT 2006 T5

With the same lawn and the same allowable mobot orientations as in the previous problem, let us call a formation "happy" if it is invariant under $120^{\circ}$ rotations. (A rotation applies both to the positions of the mobots and to their orientations.) An example of a happy formation for $n=2$ might be

Find the number of happy formations for a given $n$.