A368144 Number of ways of tiling the n X n torus up to 90-degree rotations of the square by a tile that is fixed only under 180-degree rotation of the square.

1, 4, 24, 1155, 337600, 477339104, 2872202028544, 72057595967327280, 7462505059899321934848, 3169126500571074529208754688, 5492677668532710795071525279789056, 38716571525226776289479030777851808143360, 1106936151351216411420552029913564174524281470976

Offset

1,2

Links

Table of n, a(n) for n=1..13.

Peter Kagey, Illustration of a(3)=24

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-25.

Mathematica

A368144[n_] := 1/(4 n^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), 7/4*2^(n^2/2) + 2^(n^2/4)])

Crossrefs

Cf. A367531, A368137, A368141, A368143, A368145.

Sequence in context: A193484 A024252 A368141 * A303131 A012124 A108185

Adjacent sequences: A368141 A368142 A368143 * A368145 A368146 A368147

Keyword

nonn

Author

Peter Kagey, Dec 16 2023

Status

approved