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]]])
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.
Original entry on oeis.org
2, 6, 36, 1282, 340880, 477513804, 2872221202512, 72057600262282324, 7462505061854009276768, 3169126500572875969052992416, 5492677668532714149024993226980288, 38716571525226776302072008065489884436832, 1106936151351216411420647256070432280699273711360
Offset: 1
- 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.
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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))]]])
A368140
Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by a tile that is fixed under only diagonal reflection.
Original entry on oeis.org
1, 4, 22, 1154, 337192, 477360876, 2872203226920, 72057597041056852, 7462505060326909791920, 3169126500571693774150807456, 5492677668532711895587506949961184, 38716571525226776294594927800946276718944, 1106936151351216411420589971585441310578379941760
Offset: 1
- Peter Kagey, Illustration of a(3)=22
- 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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A368140[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[EvenQ[n], (3*2^(n^2/2 - 2)), 0] + n*DivisorSum[n, Function[d, EulerPhi[d] If[EvenQ[d], 2^(n^2/(2 d) + 1), 2^((n^2 + n)/(2d))]]])
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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