A367527 -id:A367527 - 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).

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.

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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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.

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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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Mathematica

Formula