A367532 -id:A367532 - cp's OEIS frontend

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.

1, 6, 136, 16456, 8390656, 17179934976, 140737496743936, 4611686019501162496, 604462909807864343166976, 316912650057057631849169289216, 664613997892457937028364282443595776, 5575186299632655785385110159782842147536896, 187072209578355573530071668259090783432992763150336

Offset: 1

Peter Kagey, Dec 11 2023

nonn

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.

Cf. A047937, A367532.

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

a(2*n-1) = 2^(2n^2 - 4n - 1)*(4^n + 4^n^2).

a(2*n) = 2^(n^2 - 2)*(2 + 2^n^2 + 8^n^2).

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

Peter Kagey, Dec 16 2023

nonn

M.C. Escher enumerated a(2) = 23 by hand in May 1942, being perhaps the first person to attempt this sort of counting problem. (See Doris Schattschneider's book in the references for more details.)

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.

Cf. A367530, A367532, A368138, A368142, A368143, A368144.

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)])

cp's OEIS Frontend

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.

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