A255015 Number of toroidal n X n binary arrays, allowing rotation of rows and/or columns as well as matrix transposition.

1, 2, 6, 44, 2209, 674384, 954623404, 5744406453840, 144115192471496836, 14925010120653819583840, 6338253001142965335834871200, 10985355337065423791175013899922368, 77433143050453552587418968170813573149024

Offset: 0

Author

Stewart N. Ethier, Feb 12 2015

Keywords

• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..57
• 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.
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.
• Wikipedia, Polya enumeration theorem.

Crossrefs

Cf. A184271 (number of m X n binary arrays allowing rotation of rows/columns), A179043 (main diagonal of A184271), A222188 (number of m X n binary arrays allowing rotation/reflection of rows/columns), A209251 (main diagonal of A222188), A255016 (number of n X n binary arrays allowing rotation/reflection of rows/columns as well as matrix transposition).

Programs

Mathematica

a[n_] := (2 n^2)^(-1) 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}] + (2 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n (n + d - 2 IntegerPart[d/2])/(2 d)), 0], {d, 1, n}];

Formula

a(n) = (2*n^2)^{-1} Sum_{ c divides n } Sum_{ d divides n } phi(c)*phi(d)* 2^(n^2/lcm(c,d)) + (2*n)^{-1} Sum_{ d divides n } phi(d)*2^(n*(n + d - 2 *floor(d/2))/(2*d)), where phi is A000010.

Extensions

a(0)=1 from Alois P. Heinz, Feb 19 2015