A255016 Number of toroidal n X n binary arrays, allowing rotation and/or reflection of rows and/or columns as well as matrix transposition.

1, 2, 6, 26, 805, 172112, 239123150, 1436120190288, 36028817512382026, 3731252531904348833632, 1584563250300891724601560272, 2746338834266358751489231123956672, 19358285762613388352671214587818634041520

Offset

0,2

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:1502.03792 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015) # 15.8.3.

S. N. Ethier and Jiyeon Lee, Parrondo games with two-dimensional spatial dependence, arXiv preprint arXiv:1510.06947 [math.PR], 2015.

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, Pólya enumeration theorem.

Mathematica

a[n_] := (8 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}] + (4 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n^2/d), 0], {d, 1, n}] + If[Mod[n, 2] == 1, (4 n)^(-1) Sum[If[Mod[n, d] == 0 && Mod[d, 2] == 1, EulerPhi[d] (2^(n (n + 1)/(2 d)) - 2^(n^2/d)), 0], {d, 1, n}], (8 n)^(-1) Sum[If[Mod[n, d] == 0 && Mod[d, 2] == 1, EulerPhi[d] (2^(n^2/(2 d)) + 2^(n (n + 2)/(2 d)) - 2 2^(n^2/d)), 0], {d, 1, n}]] + (1/2) If[Mod[n, 2] == 1, 2^((n^2 - 3)/2), 7 2^(n^2/2 - 4)] + (4 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n (n + d - 2 IntegerPart[d/2])/(2 d)), 0], {d, 1, n}] + If[Mod[n, 2] == 1, 2^((n^2 - 5)/4), 5 2^(n^2/4 - 3)];

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), A255015 (number of n X n binary arrays allowing rotation of rows/columns as well as matrix transposition).

Cf. A054247.

Sequence in context: A331937 A162438 A137071 * A331751 A034565 A053687

Adjacent sequences: A255013 A255014 A255015 * A255017 A255018 A255019

Keyword

nonn

Author

Jiyeon Lee, Feb 12 2015

Extensions

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

Status

approved