OFFSET

0,2

COMMENTS

Consider an n X n checkerboard whose tiles are assigned colors 0 and 1, at random. There are 2^(n^2) such checkerboards. We identify the opposite edges of each checkerboard, thus making it into a (topological) torus. There are a(n) such (distinct) tori. It is possible to show that a(n) >= 2^(n^2)/n^2 for all n.

Main diagonal of A184271.

Main diagonal of Table 3: The number a(m, n) of toroidal m X n binary arrays, allowing rotation of the rows and/or the columns but not reflection, for m, n = 1, 2, ..., 8, at page 5 of Ethier. - Jonathan Vos Post, Jan 14 2013

This is a 2-dimensional generalization of binary necklaces (A000031). - Gus Wiseman, Feb 04 2019

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..57

S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352v1 [math.CO], Jan 10, 2013 and J. Int. Seq. 16 (2013) #13.4.7 .

S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015) # 15.8.3.

Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.

Peter Kagey and William Keehn, Counting Tilings of the n X m Grid, Cylinder, and Torus, J. Int. Seq. (2024) Vol. 27, Art. No. 24.6.1. See p. 2.

Wikipedia, Pólya enumeration theorem

FORMULA

a(n) = (1/n^2)*Sum_{ c divides n } Sum_{ d divides n } phi(c)*phi(d)*2^(n^2/lcm(c,d)), where phi is A000010 and lcm stands for least common multiple. - Stewart N. Ethier, Aug 24 2012

EXAMPLE

From Gus Wiseman, Feb 04 2019: (Start)

Inequivalent representatives of the a(2) = 7 checkered tori:

[0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [1 1]

[0 0] [0 1] [1 1] [0 1] [1 0] [1 1] [1 1]

(End)

MATHEMATICA

a[n_] := Sum[If[Mod[n, c] == 0, Sum[If[Mod[n, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d]), 0], {d, 1, n}], 0], {c, 1, n}]/n ^2

CROSSREFS

KEYWORD

nonn

AUTHOR

Rouben Rostamian (rostamian(AT)umbc.edu), Jun 25 2010

EXTENSIONS

Terms a(6) and a(7) from A184271

a(8)-a(12) from Stewart N. Ethier, Aug 24 2012

a(0)=1 prepended by Alois P. Heinz, Aug 20 2017

STATUS

approved