A368139 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.

2, 6, 36, 1282, 340880, 477513804, 2872221202512, 72057600262282324, 7462505061854009276768, 3169126500572875969052992416, 5492677668532714149024993226980288, 38716571525226776302072008065489884436832, 1106936151351216411420647256070432280699273711360

Offset: 1

Author

Peter Kagey, Dec 16 2023

Keywords

• nonn

Links

• 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).
• Peter Kagey, Illustration of a(3)=36
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023. See also J. Int. Seq., (2024) Vol. 27, Art. No. 24.6.1, pp. A-21, A-23.

Crossrefs

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

Programs

Mathematica

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))]]])