This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A179043 #49 Jul 05 2024 11:16:50
%S A179043 1,2,7,64,4156,1342208,1908897152,11488774559744,288230376353050816,
%T A179043 29850020237398264483840,12676506002282327791964489728,
%U A179043 21970710674130840874443091905462272,154866286100907105149651981766316633972736
%N A179043 Number of n X n checkered tori.
%C A179043 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.
%C A179043 Main diagonal of A184271.
%C A179043 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
%C A179043 This is a 2-dimensional generalization of binary necklaces (A000031). - _Gus Wiseman_, Feb 04 2019
%H A179043 Alois P. Heinz, <a href="/A179043/b179043.txt">Table of n, a(n) for n = 0..57</a>
%H A179043 S. N. Ethier, <a href="http://arxiv.org/abs/1301.2352">Counting toroidal binary arrays</a>, arXiv:1301.2352v1 [math.CO], Jan 10, 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Ethier/ethier2.html">J. Int. Seq. 16 (2013) #13.4.7</a> .
%H A179043 S. N. Ethier and Jiyeon Lee, <a href="http://arxiv.org/abs/1502.03792">Counting toroidal binary arrays, II</a>, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Lee/lee6.html">J. Int. Seq. 18 (2015) # 15.8.3</a>.
%H A179043 Veronika Irvine, <a href="http://hdl.handle.net/1828/7495">Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns</a>, PhD Dissertation, University of Victoria, 2016.
%H A179043 Peter Kagey and William Keehn, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kagey/kagey6.html">Counting Tilings of the n X m Grid, Cylinder, and Torus</a>, J. Int. Seq. (2024) Vol. 27, Art. No. 24.6.1. See p. 2.
%H A179043 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pólya_enumeration_theorem">Pólya enumeration theorem</a>
%F A179043 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
%e A179043 From _Gus Wiseman_, Feb 04 2019: (Start)
%e A179043 Inequivalent representatives of the a(2) = 7 checkered tori:
%e A179043 [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [1 1]
%e A179043 [0 0] [0 1] [1 1] [0 1] [1 0] [1 1] [1 1]
%e A179043 (End)
%t A179043 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
%Y A179043 Cf. A184271 (n X k toroidal binary arrays).
%Y A179043 Cf. A000031, A001037, A008965.
%Y A179043 Cf. A323858, A323859, A323861, A323863, A323865, A323870, A323872.
%K A179043 nonn
%O A179043 0,2
%A A179043 Rouben Rostamian (rostamian(AT)umbc.edu), Jun 25 2010
%E A179043 Terms a(6) and a(7) from A184271
%E A179043 a(8)-a(12) from _Stewart N. Ethier_, Aug 24 2012
%E A179043 a(0)=1 prepended by _Alois P. Heinz_, Aug 20 2017