A295229 -id:A295229 - cp's OEIS frontend

A295223 Number of tilings of the n X n torus, using diagonal lines to connect the gridpoints.

1, 4, 18, 669, 170440, 238773358, 1436110601256, 36028800332480074, 3731252530927004638384, 1584563250286480205777197264, 2746338834266357074512496613490144, 19358285762613388151183577985346072926384, 553468075675608205710323628035216140349636855680

Offset: 1

Peter Kagey, Nov 17 2017

nonn

			For n = 3, the following four tilings are considered equivalent:
*---*->-+---+   +---+->-*---*   *---*->-+---+   +---+->-+---+
| / | \ | \ |   | / | / | \ |   | \ | / | / |   | / | \ | \ |
*---*---+---+   +---+---*---*   *---*---+---+   *---*---+---+
^ / | / | \ ^ = ^ / | \ | \ ^ = ^ \ | / | \ ^ = ^ \ | / | / ^
+---+---+---+   +---+---+---+   +---+---+---+   *---*---+---+
| \ | / | / |   | \ | \ | / |   | / | \ | \ |   | \ | / | \ |
+---+->-+---+   +---+->-+---+   +---+->-+---+   +---+->-+---+
The transformations are horizontal reflection, shifting to the right, and shifting down.

Peter Kagey, Table of n, a(n) for n = 1..57
Peter Kagey, The eighteen tilings of the 3 x 3 torus.
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A255016, A295229.

a[n_] := 1/(8*n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + If[OddQ[n], n^2*2^((n^2 + 1)/2), n^2/4 (3*2^(n^2/2) + 2^((n^2 + 4)/2))] + 2*If[EvenQ[n], n/2*DivisorSum[n, Function[c, EulerPhi[c] (2^(n^2/LCM[2, c]) + If[OddQ[c], 0, 2^(n^2/c)])]], n*DivisorSum[n, Function[c, EulerPhi[c]*If[OddQ[c], 0, 2^(n^2/c)]]]] + If[OddQ[n], 0, n^2 (2^(n^2/4))] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 2^((n^2 + n)/(2 d)), EvenQ[d], 2^(n^2/(2 d))]]])

A367524 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under horizontal reflection, but no other symmetries of the square.

Original entry on oeis.org

1, 39, 32896, 536895552, 140737496743936, 590295810384475521024, 39614081257132309534260330496, 42535295865117307939839354957685850112, 730750818665451459101843020821051317142553624576, 200867255532373784442745261543120694290360960529885344825344

Offset: 1

Peter Kagey, Dec 10 2023

nonn

Also, this is the number ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under 180-degree rotation, but no other symmetries of the square.

Peter Kagey, Illustration of a(2)=39
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-7, A-8.

Cf. A054247, A295229, A302484, A367522, A367523, A367525.

Table[{2^(4 m^2 - 4 m - 2) (2 + 2^(2 m - 1)^2), 2^(2 m^2 - 3) (2 + 3*4^m^2 + 64^m^2)}, {m, 1, 5}] // Flatten

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

a(2m) = 2^(2m^2 - 3)*(2 + 3*4^m^2 + 64^m^2).

A367525 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is not fixed under any of the symmetries of the square.

Original entry on oeis.org

1, 538, 16777216, 35184378381312, 4722366482869645213696, 40564819207303347603293977182208, 22300745198530623141535718272648361505980416, 784637716923335095479473677930668862955643627524327473152, 1766847064778384329583297500742918515827483896875618958121606201292619776

Offset: 1

Peter Kagey, Dec 10 2023

nonn

Peter Kagey, Illustration of a(2)=538
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-8.

Cf. A054247, A295229, A302484, A367522, A367523, A367524.

Table[{4096^(m^2 - m), 8^(m^2 - 1) (512^m^2 + 3*8^m^2 + 2)}, {m, 1, 5}] // Flatten

a(2m-1) = 4096^(m^2 - m).

a(2m) = 8^(m^2 - 1)*(512^m^2 + 3*8^m^2 + 2).

A367522 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under both horizontal and vertical reflection, but not diagonal reflection.

Original entry on oeis.org

1, 4, 84, 8292, 4203520, 8590033024, 70368815480832, 2305843010824323072, 302231454912728264605696, 158456325028529097399561355264, 332306998946228986960926214931349504, 2787593149816327892693735671512138485071872, 93536104789177786765036453099565034406633831137280

Offset: 1

Peter Kagey, Nov 21 2023

nonn

The a(2) = 4 tilings are
  - -   - -   - |       - |
  - -,  | -,  - |, and  | -.

Michael De Vlieger, Table of n, a(n) for n = 1..57
Peter Kagey, Illustration of a(3)=84.
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, p. A-6.

Cf. A054247, A295229, A302484, A367523, A367524, A367525.

a[n_] := If[EvenQ[n], 2^(#^2 - 3)*(2 + 3*2^#^2 + 8^#^2) &[n/2], 4^(#^2 - 2 # - 1)*(4^# + 4^#^2 + 8^#) &[(n + 1)/2]]; Array[a, 13] (* Michael De Vlieger, Jul 06 2024 *)

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

a(2m) = 2^(m^2 - 3)*(2 + 3*2^m^2 + 8^m^2).

A367523 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under 90-degree rotations, but not reflections.

Original entry on oeis.org

1, 4, 70, 8292, 4195360, 8590033024, 70368748374016, 2305843010824323072, 302231454903932172107776, 158456325028529097399561355264, 332306998946228968514182141758668800, 2787593149816327892693735671512138485071872, 93536104789177786765035834129545391718695404830720

Offset: 1

Peter Kagey, Dec 10 2023

nonn

Peter Kagey, Illustration of a(3)=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-7.

Cf. A054247, A295229, A302484, A367522, A367524, A367525.

Table[{2^(-2 + (-4 + n) n) (2^(n (2 + n)) + 2^(1 + 3 n) + 8^n^2), 2^(-3 + n^2) (2 + 3 2^n^2 + 8^n^2)}, {n, 1, 5}] // Flatten

a(2m-1) = 2^(m^2 - 4m - 2)*(2^(3m+1) + 2^(m^2+2m) + 8^m^2).

a(2m) = 2^(m^2 - 3)*(2 + 3*2^m^2 + 8^m^2) = A367522(2m).

cp's OEIS Frontend

A295223 Number of tilings of the n X n torus, using diagonal lines to connect the gridpoints.

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A367524 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under horizontal reflection, but no other symmetries of the square.

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A367525 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is not fixed under any of the symmetries of the square.

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A367522 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under both horizontal and vertical reflection, but not diagonal reflection.

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A367523 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under 90-degree rotations, but not reflections.

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