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.

1, 83, 1452, 1770, 15587, 19863, 131980, 169716, 182884, 971013, 1245461, 1389317, 6508358, 8289158, 9408838, 9790598, 40813063, 51522567, 58997063, 62834759, 243405576, 304396296, 349949576, 378076936, 387585288

Offset: 2

Author

L. Edson Jeffery, Dec 12 2013

Keywords

• nonn
• tabf

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 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.)

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?

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?

The problems are very difficult, and here A233332 gives a solution only for the simplest case m=1.

Examples

			Array begins:
...........1
..........83
........1452.........1770
.......15587........19863
......131980.......169716.......182884
......971013......1245461......1389317
.....6508358......8289158......9408838......9790598
....40813063.....51522567.....58997063.....62834759
...243405576....304396296....349949576....378076936....387585288
..1395618313...1728983049...1990082057...2169422089...2260674313
..7751398922...9515886602..10947167754..12001065994..12646026762..12863117322
.41932226571..51033062411..58616206347..64480008203..68473230347..70495047691
		

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-A233333.

Programs

Mathematica

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 *)

Formula

The Jeffery PDF contains an algorithm for constructing this array.

Conjecture: For all n and for all k, A(n,k) == n-2 (mod 2^(n-3)).