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.
KEYWORD
nonn
AUTHOR
Jiyeon Lee, Feb 12 2015
EXTENSIONS
a(0)=1 from Alois P. Heinz, Feb 19 2015
STATUS
approved