A368141
Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflections of the square by a tile that is fixed under only 180-degree rotations.
Original entry on oeis.org
1, 4, 24, 1154, 337600, 477339020, 2872202028544, 72057595967315028, 7462505059899321934848, 3169126500571074529202043808, 5492677668532710795071525279789056, 38716571525226776289479030777837491607904, 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-24.
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A368141[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[EvenQ[n], (7*2^((n^2 - 4)/2)), 2^((n^2 + 1)/2)] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 2^(n^2/(2 d)), 0]]])
A368138
Number of ways of tiling the n X n torus up to the symmetries of the square by an asymmetric tile.
Original entry on oeis.org
1, 154, 1864192, 2199026796168, 188894659314785812480, 1126800533536206914843196839296, 455117248949604553908892209645884928950272, 12259964326927110866866776228808161337250421224373748224, 21812926725659065797324660502998994022561529591086874194578215566049280
Offset: 1
- Dan Davis, On a Tiling Scheme from M. C. Escher, Electron. J. Combin. 4(2) (1996), #R23.
- Peter Kagey, Illustration of a(2)=154
- 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, p. A-23.
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A368138[n_] := 1/(8n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 8^(n^2/LCM[c, d])]]]] + If[EvenQ[n], n^2 (3/4*8^(n^2/2) + 8^(n^2/4)) + n*DivisorSum[n, Function[c, EulerPhi[c] (If[EvenQ[c], 2*8^(n^2/c), 8^(n^2/(2 c))])]], 0] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 8^(n^2/(2 d)), 0]]])
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)])
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