

A179043


Number of n X n checkered tori.


15



1, 2, 7, 64, 4156, 1342208, 1908897152, 11488774559744, 288230376353050816, 29850020237398264483840, 12676506002282327791964489728, 21970710674130840874443091905462272, 154866286100907105149651981766316633972736
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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 2dimensional 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.
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

Cf. A184271 (n X k toroidal binary arrays).
Cf. A000031, A001037, A008965.
Cf. A323858, A323859, A323861, A323863, A323865, A323870, A323872.
Sequence in context: A011821 A117263 A046855 * A116985 A042051 A196925
Adjacent sequences: A179040 A179041 A179042 * A179044 A179045 A179046


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



