Tough enumeration problem from the theory of tiles.

[L. Edson Jeffery]

Hello,

Are there any tiling theorists out there?

I recently submitted to OEIS the irregular array A233332. This array gives the number of first coronas of a fixed rhombus (with characteristics of n-fold rotational symmetry, n>=2) in the Euclidean plane. I worked out an algorithm to generate the terms since no formula is known. Generating functions for the columns are not known either. Evidently the more general problem in tiling theory of counting the number of possible N-th coronas of these rhombi is an open one. For N>1, I think this problem is very hard.

The difficulty I am having is with the reduction of A233332 for symmetry. This reduced version is supposed to be A233333 in which rotations and reflections are not counted as distinct.

I have done most of the work on the algorithm for A233333 but hit a snag only when counting coronas that have a dihedral symmetry or symmetry properties of a swastika. Consequently, the four terms (totaling three rows) shown in A233333 (draft) so far were counted essentially by hand. It is getting to be too difficult to continue counting this way.

So, I need help with A233333, if anyone is willing. The snag mentioned above involves some kind of combination of counting partitions and compositions (either sums or products of them, or both) which continues to elude me.

I uploaded a PDF to A233332 that describes the algorithm for constructing A233332. For anyone interested in helping me, please read that document first, and then contact me privately to discuss A233333, unless it is alright to continue the dialog here.

Thank you in advance,

Ed Jeffery