A233333 Irregular array read by rows: A(n,k) = number of first coronas of a fixed rhombus r_{n,k} with characteristics of n-fold rotational symmetry in the Euclidean plane, n>=2, 1<=k<=floor(n/2), reduced for symmetry, as explained below.

1, 28, 414, 247

Offset: 2

Author

L. Edson Jeffery, Dec 07 2013

Keywords

• nonn
• tabf
• hard
• more

Comments

Row index n begins with 2, column index k begins with 1.

Let R_n be the set of floor(n/2) rhombi in which the k-th rhombus r_{n,k} has interior angles about its vertices (or corners) given by the pair (k*Pi/n, (n-k)*Pi/n), n>=2, 1<=k<=floor(n/2). Let T be any tiling of the plane. For any tile t in T, let C_m(t) denote the m-th corona of t, m>=0. Equivalently, starting with any tile r in R_n fixed in the plane, we can compose a corona C_m(r) of r of any order m by tessellation using tiles of R_n. This leads to the following problem in the theory of tiles and its reduction for symmetry which seem to have not been addressed before in the literature. (See [Jeffery] for details and definitions.)

Problem: For r_{n,k} in R_n fixed in the plane, in how many ways can r_{n,k} be extended to an m-th corona of r_{n,k} using tiles of R_n?

Array A233332 gives a solution for the case m=1. Here A233333 gives a solution for m=1 when rotations and reflections are not counted.

Examples

			Array begins:
1;
28;
414, 247;
...
		

References

• Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995, p. 145.

Links

• Dirk Frettlöh, Glossary of tiling terms, Tilings Encyclopedia.
• L. E. Jeffery, Constructing A233332.
• Eric W. Weisstein, Corona, from MathWorld.
• Eric W. Weisstein, Tiling, from MathWorld.

Crossrefs

Cf. A233329-A233332.