%I #14 Sep 24 2017 12:15:33

%S 1,7,21,15,51,66,102,136,75,186,244,274,310,406,462,246,490,631,729,

%T 780,735,939,1086,1183,610,1065,1341,1557,1707,1783,1491,1861,2155,

%U 2380,2511,1286,2037,2512,2908,3217,3427,3534,2716,3322,3831,4250,4548,4738,2404

%N Irregular array read by rows: T(n,k) = number of r_{n,k}-cores associated with A233332(n,k), reduced for symmetry, for n>=2, 1<=k<=floor(n/2), explained below.

%C For definitions and more details, see the PDF by L. E. Jeffery.

%C Let n be an integer, n >= 2, and let k in {1,...,floor(n/2)}. Let R_n be the set of floor(n/2) rhombi in which the k-th rhombus r_{n,k} in R_n has interior angles about its vertices (or corners) given by the pair (k*Pi/n, (n-k)*Pi/n). 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 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. For any r in R_n fixed in the plane, a disjoint union of r with four tiles t_1,t_2,t_3,t_4 in R_n is called a "candidate." If no tiles overlap in a candidate, then that candidate is called an "r-core;" otherwise that candidate is rejected (since no corona of r can be constructed from it). For each r_{n,k} in R_n, and for m>0, every r_{n,k}-core can be extended to an m-th corona of r_{n,k} using tiles of R_n in a number of ways. For the case m=1, the array A233332 gives the number of ways that this can be done for each n and k. In the theory of tiles the general problem of counting these coronas and its reduction for symmetry seem to have not been addressed before in the literature.

%C The array A233330 gives the number of r_{n,k}-cores associated with the coronas enumerated in A233332. The present array A233331 gives the number of r_{n,k}-cores associated with the coronas enumerated in A233332 when isometries different from the identity are not counted.

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

%H Dirk Frettlöh, <a href="http://tilings.math.uni-bielefeld.de/glossary">Glossary of tiling-theoretic terms</a>, Tilings Encyclopedia.

%H L. E. Jeffery, <a href="/A233332/a233332_5.pdf">Algorithm for constructing A233332</a>.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Corona.html">Corona</a>, from MathWorld.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Tiling.html">Tiling</a>, from MathWorld.

%F Conjecture: G.f. for column 1 is F_1(x) = x^2*(1+4x+x^2)/((1+x)^2*(1-x)^5).

%F Conjecture: Column 1 = A233329, up to an offset.

%F Conjecture: Column k has generating function of the form F_k(x) = G_k(x)/((1+x)^2*(1-x)^5), where G_k(x) is a polynomial in x.

%F Conjecture: A233331(2*M,1) = A076454(M), M>=1.

%e Array begins: {1}; {7}; {21, 15}; {51, 66}; {102, 136, 75}; ...

%Y Cf. A076454, A233329-A233333.

%K nonn,tabf

%O 2,2

%A _L. Edson Jeffery_, Jan 06 2014