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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.

%I A233333 #29 Feb 16 2025 08:33:21
%S A233333 1,28,414,247
%N 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.
%C A233333 Row index n begins with 2, column index k begins with 1.
%C A233333 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.)
%C A233333 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?
%C A233333 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.
%D A233333 Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995, p. 145.
%H A233333 Dirk Frettlöh, <a href="http://tilings.math.uni-bielefeld.de/glossary">Glossary of tiling terms</a>, Tilings Encyclopedia.
%H A233333 L. E. Jeffery, <a href="/A233332/a233332_4.pdf">Constructing A233332</a>.
%H A233333 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Corona.html">Corona</a>, from MathWorld.
%H A233333 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Tiling.html">Tiling</a>, from MathWorld.
%e A233333 Array begins:
%e A233333 1;
%e A233333 28;
%e A233333 414, 247;
%e A233333 ...
%Y A233333 Cf. A233329-A233332.
%K A233333 nonn,tabf,hard,more
%O A233333 2,2
%A A233333 _L. Edson Jeffery_, Dec 07 2013