This seems like a specific example of a more general question. Let S be the permutation group on an n-element set X, and let H be a subgroup of S. Any element of S induces a map from graphs with vertices in X to other maps with vertices in X. Set two graphs equivalent if they are mapped to each other by elements of H, and count the equivalence classes produced by that relation. The problem you're studying is the example of this where H is the dihedral group (rotations and reflections).

The general question seems like a very natural thing to study, and the dihedral group is one of the most natural subgroups to use (along with, say, the cyclic group and the alternating group), so I would have guessed this would already be in the literature. But the sequence of numbers you generated doesn't appear in Sloane's integer sequence database, so perhaps this is unexplored territory.

You could also break the problem down further looking for numbers of the form F(n,k), where k is the number of edges in the graph. I'd be curious what, for example, the sequence F(n,n) looks like.