

A233332


Irregular array read by rows: A(n,k) = number of first coronas of a fixed rhombus r_{n,k} with characteristics of nfold rotational symmetry in the Euclidean plane, n>=2, 1<=k<=floor(n/2), as explained below.


6



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
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OFFSET

2,2


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 kth rhombus r_{n,k} has interior angles about its vertices (or corners) given by the pair (k*Pi/n, (nk)*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 mth 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 mth 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.


REFERENCES

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


LINKS

Table of n, a(n) for n=2..26.
Dirk FrettlĂ¶h, Glossary of tiling terms, Tilings Encyclopedia.
L. E. Jeffery, Constructing A233332.
Eric W. Weisstein, Corona, from MathWorld.
Eric W. Weisstein, Tiling, from MathWorld.


FORMULA

The Jeffery PDF contains an algorithm for constructing this array.
Conjecture: For all n and for all k, A(n,k) == n2 (mod 2^(n3)).


EXAMPLE

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


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 *)


CROSSREFS

Cf. A233329A233333.
Sequence in context: A250083 A292284 A195893 * A175662 A103233 A093283
Adjacent sequences: A233329 A233330 A233331 * A233333 A233334 A233335


KEYWORD

nonn,tabf


AUTHOR

L. Edson Jeffery, Dec 12 2013


STATUS

approved



