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A233330 Irregular array read by rows: T(n,k) = number of r_{n,k}-cores associated with A233332(n,k), for n>=2, 1<=k<=floor(n/2), explained below.

%I A233330 #20 Feb 16 2025 08:33:21
%S A233330 1,16,64,81,169,225,361,484,527,676,900,1018,1156,1521,1752,1831,1849,
%T A233330 2401,2789,2989,2809,3600,4195,4564,4689,4096,5184,6042,6634,6936,
%U A233330 5776,7225,8408,9283,9820,10001,7921,9801,11377,12601,13437,13861,10609,12996,15039
%N A233330 Irregular array read by rows: T(n,k) = number of r_{n,k}-cores associated with A233332(n,k), for n>=2, 1<=k<=floor(n/2), explained below.
%C A233330 For definitions and more details, see the PDF by L. E. Jeffery.
%C A233330 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 A233330 The present array A233330 gives the number of r_{n,k}-cores associated with the coronas enumerated in A233332. The 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 A233330 Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995, p. 145.
%H A233330 Dirk Frettlöh, <a href="http://tilings.math.uni-bielefeld.de/glossary">Glossary of tiling-theoretic terms</a>, Tilings Encyclopedia.
%H A233330 L. E. Jeffery, <a href="/A233332/a233332_5.pdf">Algorithm for constructing A233332</a>.
%H A233330 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Corona.html">Corona</a>, from MathWorld.
%H A233330 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Tiling.html">Tiling</a>, from MathWorld.
%F A233330 The Jeffery PDF contains an algorithm for constructing this array.
%F A233330 Conjecture: The entries of column k satisfy sum{h=0..5} ((-1)^h*binomial(5,h)*A(n+5-h,k)) = 0, for appropriate initial conditions.
%e A233330 Array begins: {1}; {16}; {64, 81}; {169, 225}; {361, 484, 527}; ...
%t A233330 maxn := 13; t[n_, m_, i_] := 1 + Mod[Floor[m/(n - 1)^(4 - i)], n - 1]; e[n_, k_, m_, 1] :=  -t[n, m, 1] + k + t[n, m, 4]; e[n_, k_, m_, 2] := -t[n, m, 2] - k + n + t[n, m, 1]; e[n_, k_, m_, 3] :=  -t[n, m, 3] + k + t[n, m, 2]; e[n_, k_, m_, 4] :=  -t[n, m, 4] - k + n + t[n, m, 3]; cores[n_, k_] := Sum[Product[If[e[n, k, m, i] >= 0, 1, 0], {i, 1, 4}], {m, 0, (n - 1)^4 - 1}]; Flatten[Table[cores[n, k], {n, 2, maxn}, {k, 1, Floor[n/2]}]] (* _L. Edson Jeffery_, Jul 22 2014 *)
%Y A233330 Cf. A233329-A233333.
%K A233330 nonn,tabf
%O A233330 2,2
%A A233330 _L. Edson Jeffery_, Dec 29 2013