This site is supported by donations to The OEIS Foundation.



The submissions stack has been unacceptably high for several months now. Please voluntarily restrict your submissions and please help with the editing. (We don't want to have to impose further limits.)

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A179043 Number of n X n checkered tori. 4
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.

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 | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified August 28 13:24 EDT 2015. Contains 261122 sequences.