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A233332 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), as explained below.

%I A233332 #93 Feb 16 2025 08:33:21
%S A233332 1,83,1452,1770,15587,19863,131980,169716,182884,971013,1245461,
%T A233332 1389317,6508358,8289158,9408838,9790598,40813063,51522567,58997063,
%U A233332 62834759,243405576,304396296,349949576,378076936,387585288
%N A233332 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), as explained below.
%C A233332 Row index n begins with 2, column index k begins with 1.
%C A233332 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 by tiles of R_n. 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 A233332 Problem 1: 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 A233332 Problem 2: From Problem 1, in how many ways can r_{n,k} be so extended if isometries different from the identity are not counted?
%C A233332 The problems are very difficult, and here A233332 gives a solution only for the simplest case m=1.
%D A233332 Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995, p. 145.
%H A233332 Dirk Frettlöh, <a href="http://tilings.math.uni-bielefeld.de/glossary">Glossary of tiling terms</a>, Tilings Encyclopedia.
%H A233332 L. E. Jeffery, <a href="/A233332/a233332_5.pdf">Constructing A233332</a>.
%H A233332 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Corona.html">Corona</a>, from MathWorld.
%H A233332 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Tiling.html">Tiling</a>, from MathWorld.
%F A233332 The Jeffery PDF contains an algorithm for constructing this array.
%F A233332 Conjecture: For all n and for all k, A(n,k) == n-2 (mod 2^(n-3)).
%e A233332 Array begins:
%e A233332 ...........1
%e A233332 ..........83
%e A233332 ........1452.........1770
%e A233332 .......15587........19863
%e A233332 ......131980.......169716.......182884
%e A233332 ......971013......1245461......1389317
%e A233332 .....6508358......8289158......9408838......9790598
%e A233332 ....40813063.....51522567.....58997063.....62834759
%e A233332 ...243405576....304396296....349949576....378076936....387585288
%e A233332 ..1395618313...1728983049...1990082057...2169422089...2260674313
%e A233332 ..7751398922...9515886602..10947167754..12001065994..12646026762..12863117322
%e A233332 .41932226571..51033062411..58616206347..64480008203..68473230347..70495047691
%t A233332 maxn := 10; t[n_, m_, i_] := 1 + Mod[Floor[m/(n - 1)^(4 - i - 1)], n - 1]; e[n_, k_, m_, i_] := -t[n, m, i] + (-1)^(i)*k + Mod[i, 2]*n + t[n, m, Mod[i - 1, 4]]; a[n_, k_] := Sum[Product[If[e[n, k, m, i] >= 0, 1, 0], {i, 0, 3}]*Product[SeriesCoefficient[Series[(1 - x)/(1 - 2*x + x^n), {x, 0, 2*n - k - 2}], e[n, k, m, i]], {i, 0, 3}], {m, 0, (n - 1)^4 - 1}]; Grid[Table[a[n, k], {n, 2, maxn}, {k, Floor[n/2]}]] (* _L. Edson Jeffery_, Jul 22 2014 *)
%Y A233332 Cf. A233329-A233333.
%K A233332 nonn,tabf
%O A233332 2,2
%A A233332 _L. Edson Jeffery_, Dec 12 2013