A047945
Number of 10-colorings of an n X n grid, up to rotational symmetry.
Original entry on oeis.org
1, 10, 2530, 250025500, 2500000025005000, 2500000000002500005000000, 250000000000000000250000000500000000, 2500000000000000000000002500000000005000000000000, 2500000000000000000000000000000025000000000000005000000000000000
Offset: 0
A367531
The number of ways of tiling the n X n grid up to 90-degree rotation by a tile that is fixed under 180-degree rotation but not 90-degree rotation.
Original entry on oeis.org
1, 6, 136, 16456, 8390656, 17179934976, 140737496743936, 4611686019501162496, 604462909807864343166976, 316912650057057631849169289216, 664613997892457937028364282443595776, 5575186299632655785385110159782842147536896, 187072209578355573530071668259090783432992763150336
Offset: 1
- Peter Kagey, Illustration of a(3)=136
- 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-6, A-10.
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Table[{2^(2 m^2 - 4 m - 1)*(4^m + 4^m^2), 2^(m^2 - 2)*(2 + 2^m^2 + 8^m^2)}, {m, 1, 5}] // Flatten
A367532
The number of ways of tiling the n X n grid up to 90-degree rotation by a tile that is not fixed under 180-degree rotation.
Original entry on oeis.org
1, 70, 65536, 1073758336, 281474976710656, 1180591620734591303680, 79228162514264337593543950336, 85070591730234615870455337878516924416, 1461501637330902918203684832716283019655932542976, 401734511064747568885490523085607563280607806359022338048000
Offset: 1
- Peter Kagey, Illustration of a(2)=70
- 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-6, A-10.
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Table[{256^(m^2 - m), 4^(m^2 - 1)*(2 + 4^m^2 + 64^m^2)}, {m, 1, 5}] // Flatten
A368143
Number of ways of tiling the n X n torus up to 90-degree rotations of the square by two tiles that are each fixed under 90-degree rotation of the square.
Original entry on oeis.org
2, 6, 28, 1171, 337664, 477339616, 2872202032640, 72057595967392816, 7462505059899322983424, 3169126500571074529242309120, 5492677668532710795071526353530880, 38716571525226776289479030777920527620096, 1106936151351216411420552029913564178922327982080
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)=28
- 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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A368143[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) + 2^((n^2 + 7)/4), 7*2^((n^2 - 4)/2) + 5*2^(n^2/4)])