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]]])
A368142
Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by an asymmetric tile.
Original entry on oeis.org
1, 23, 7296, 67124308, 11258999068672, 32794211700912270688, 1616901275801313012113145856, 1329227995784915876578744356684451904, 18043230090504974298810923860695296894480941056, 4017345110647475688854905231100098373350012274109805442048
Offset: 1
- 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-24.
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A368142[n_] := 1/(4 n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] + n^2*If[EvenQ[n], (3*2^(n^2 - 2)), 0] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 2^(n^2/d), 0]]])
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))]]])
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