A367528 -id:A367528 - 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).

A367529 The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by a tile that is not fixed under any of these symmetries.

Original entry on oeis.org

1, 68, 65536, 1073758208, 281474976710656, 1180591620734591172608, 79228162514264337593543950336, 85070591730234615870455337876369440768, 1461501637330902918203684832716283019655932542976, 401734511064747568885490523085607563280607805796072384626688

Offset: 1

Peter Kagey, Dec 10 2023

nonn

Peter Kagey, Illustration of a(2)=68
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. A367525, A367526, A367527, A367528.

Table[{256^(m^2 - m), 1/4*(16^m^2 + 256^m^2)}, {m, 1, 5}] // Flatten

a(2m-1) = 256^(m^2 - m).

a(2m) = 1/4 (16^m^2 + 256^m^2).

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, Dec 16 2023

nonn

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.

Cf. A367528, A368137, A368139, A368140, A368142.

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]]])

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).

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A367529 The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by a tile that is not fixed under any of these symmetries.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula