This site is supported by donations to The OEIS Foundation.



The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A179043 Number of n X n checkered tori. 5
2, 7, 64, 4156, 1342208, 1908897152, 11488774559744, 288230376353050816, 29850020237398264483840, 12676506002282327791964489728, 21970710674130840874443091905462272, 154866286100907105149651981766316633972736 (list; graph; refs; listen; history; text; internal format)



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


Table of n, a(n) for n=1..12.

S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352v1 [math.CO], Jan 10, 2013.

S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015.

Wikipedia, Pólya enumeration theorem


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


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


Cf. A184271 (n X k toroidal binary arrays).

Sequence in context: A011821 A117263 A046855 * A116985 A042051 A196925

Adjacent sequences:  A179040 A179041 A179042 * A179044 A179045 A179046




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


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

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



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 27 00:21 EDT 2017. Contains 287189 sequences.