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%I #29 Nov 30 2023 12:26:27
%S 1,2,6,26,805,172112,239123150,1436120190288,36028817512382026,
%T 3731252531904348833632,1584563250300891724601560272,
%U 2746338834266358751489231123956672,19358285762613388352671214587818634041520
%N Number of toroidal n X n binary arrays, allowing rotation and/or reflection of rows and/or columns as well as matrix transposition.
%H Michael De Vlieger, <a href="/A255016/b255016.txt">Table of n, a(n) for n = 0..57</a>
%H S. N. Ethier and Jiyeon Lee, <a href="http://arxiv.org/abs/1502.03792">Counting toroidal binary arrays, II</a>, arXiv:1502.03792 [math.CO], Feb 12, 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Lee/lee6.html">J. Int. Seq. 18 (2015) # 15.8.3</a>.
%H S. N. Ethier and Jiyeon Lee, <a href="http://arxiv.org/abs/1510.06947">Parrondo games with two-dimensional spatial dependence</a>, arXiv preprint arXiv:1510.06947 [math.PR], 2015.
%H Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv:2311.13072 [math.CO], 2023. See p. 3.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pólya_enumeration_theorem">Pólya enumeration theorem</a>.
%t 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)];
%Y 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).
%Y Cf. A054247.
%K nonn
%O 0,2
%A _Jiyeon Lee_, Feb 12 2015
%E a(0)=1 from _Alois P. Heinz_, Feb 19 2015
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