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.
Original entry on oeis.org
1, 4, 24, 1155, 337600, 477339104, 2872202028544, 72057595967327280, 7462505059899321934848, 3169126500571074529208754688, 5492677668532710795071525279789056, 38716571525226776289479030777851808143360, 1106936151351216411420552029913564174524281470976
Offset: 1
- 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.
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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)])
A368145
Number of ways of tiling the n X n torus up to 90-degree rotations of the square by an asymmetric tile.
Original entry on oeis.org
1, 23, 7296, 67124336, 11258999068672, 32794211700912314368, 1616901275801313012113145856, 1329227995784915876578744357489750016, 18043230090504974298810923860695296894480941056, 4017345110647475688854905231100098373350012499289786810368
Offset: 1
- Doris Schattschneider, Visions of Symmetry, W.H. Freeman, 1990, pages 44-48.
- Peter Kagey, Illustration of a(2)=23
- 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.
- Doris Schattschneider, Escher's combinatorial patterns, Electron. J. Combin. 4(2) (1996), #R17.
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A368145[n_] := 1/(4n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] + n^2*If[OddQ[n], 0, 3/4*2^n^2 + 2^(n^2/2)])
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