A367529 -id:A367529 - cp's OEIS frontend

A367526 The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by two tiles that are each fixed under both of these reflections.

2, 9, 168, 16960, 8407040, 17180983296, 140737630961664, 4611686053860868096, 604462909825456529211392, 316912650057075646247661993984, 664613997892457973921852429862699008, 5575186299632655785536225887234636434636800, 187072209578355573530072906199130068813267662274560

Offset: 1

Peter Kagey, Dec 10 2023

nonn

Peter Kagey, Illustration of a(2)=9
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-9.

Cf. A054247, A367526, A367527, A367528, A367529.

Table[{2^(2 m^2 - 4 m - 1) (4^m + 4^m^2 + 8^m), 4^(m^2 - 1) (1 + 2^(1 + m) + 4^m^2)}, {m, 1, 5}] // Flatten

a(2m-1) = 2^(2m^2 - 4m - 1)(4^m + 4^m^2 + 8^m).

a(2m) = 4^(m^2 - 1)(1 + 2^(1 + m) + 4^m^2).

A367528 The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by a tile that is fixed under 180-degree rotations but is not fixed under either reflection.

Original entry on oeis.org

1, 5, 136, 16448, 8390656, 17179934720, 140737496743936, 4611686019501129728, 604462909807864343166976, 316912650057057631849152512000, 664613997892457937028364282443595776, 5575186299632655785385110159782807787798528, 187072209578355573530071668259090783432992763150336

Offset: 1

Peter Kagey, Dec 10 2023

nonn

Peter Kagey, Illustration of a(2)=5
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-9.

Cf. A367524, A367526, A367527, A367529.

Table[{2^(2 m^2 - 4 m - 1) (4^m + 4^m^2), (4^m^2 + 16^m^2)/4}, {m, 1, 5}] // Flatten

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

a(2m) = (4^m^2 + 16^m^2)/4.

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, Dec 16 2023

nonn

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.

Cf. A367529, A368138, A368139, A368140, A368141.

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]]])

A367527 The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by a tile that is fixed under diagonal reflection, but not antidiagonal reflection.

Original entry on oeis.org

1, 7, 144, 16704, 8396800, 17180459008, 140737555464192, 4611686036680998912, 604462909816110680375296, 316912650057066639048407252992, 664613997892457954898647603849723904, 5575186299632655785460668023508722111217664, 187072209578355573530072277557703869206096815063040

Offset: 1

Peter Kagey, Dec 10 2023

nonn

Peter Kagey, Illustration of a(2)=7
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-9.

Cf. A302484, A367526, A367528, A367529.

Table[{2^(2 m^2 - 4 m - 2) (2^(1 + 2 m^2) + 8^m), 4^(m^2 - 1) (1 + 2^m + 4^m^2)}, {m, 1, 5}] // Flatten

a(2m-1) = 2^(2m^2 - 4m - 2)*(2^(1 + 2 m^2) + 8^m).

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

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, Dec 11 2023

nonn

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.

Cf. A047937, A367525, A367529, A367531.

Table[{256^(m^2 - m), 4^(m^2 - 1)*(2 + 4^m^2 + 64^m^2)}, {m, 1, 5}] // Flatten

a(2*n-1) = 256^(n^2 - n).

a(2*n) = 4^(n^2 - 1)*(2 + 4^n^2 + 64^n^2).

cp's OEIS Frontend

A367526 The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by two tiles that are each fixed under both of these reflections.

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A367528 The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by a tile that is fixed under 180-degree rotations but is not fixed under either reflection.

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A368142 Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by an asymmetric tile.

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A367527 The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by a tile that is fixed under diagonal reflection, but not antidiagonal reflection.

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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.

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