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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.
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
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 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.
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.
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)).
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
KEYWORD
nonn,tabf
AUTHOR
L. Edson Jeffery, Dec 12 2013
STATUS
approved