login
Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by two tiles that are each fixed under both diagonal and antidiagonal reflection.
4

%I #17 Jul 09 2024 08:55:19

%S 2,6,36,1282,340880,477513804,2872221202512,72057600262282324,

%T 7462505061854009276768,3169126500572875969052992416,

%U 5492677668532714149024993226980288,38716571525226776302072008065489884436832,1106936151351216411420647256070432280699273711360

%N Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by two tiles that are each fixed under both diagonal and antidiagonal reflection.

%H S. N. Ethier and Jiyeon Lee, <a href="http://arxiv.org/abs/1502.03792">Counting toroidal binary arrays, II</a>, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Lee/lee6.html">J. Int. Seq. 18 (2015)</a>.

%H Peter Kagey, <a href="/A368139/a368139.pdf">Illustration of a(3)=36</a>

%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 also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kagey/kagey6.html">J. Int. Seq.</a>, (2024) Vol. 27, Art. No. 24.6.1, pp. A-21, A-23.

%t A368139[n_] := 1/(4n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + n^2*If[OddQ[n], 2^((n^2 + 1)/2), (7*2^(n^2/2 - 2))] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 2^(n^2/(2 d)), 2^((n^2 + n)/(2d))]]])

%Y Cf. A295223, A367526, A368140, A368141, A368142.

%K nonn

%O 1,1

%A _Peter Kagey_, Dec 16 2023