A367524 -id:A367524 - cp's OEIS frontend

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.

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).

A368218 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under horizontal reflection only.

Original entry on oeis.org

1, 3, 2, 4, 7, 3, 10, 20, 24, 6, 16, 76, 144, 76, 10, 36, 272, 1120, 1056, 288, 20, 64, 1072, 8448, 16576, 8320, 1072, 36, 136, 4160, 66816, 262656, 263680, 65792, 4224, 72, 256, 16576, 528384, 4197376, 8396800, 4197376, 525312, 16576, 136

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k |  1    2     3       4         5           6
  ----+--------------------------------------------
    1 |  1    3     4      10        16          36
    2 |  2    7    20      76       272        1072
    3 |  3   24   144    1120      8448       66816
    4 |  6   76  1056   16576    262656     4197376
    5 | 10  288  8320  263680   8396800   268517376
    6 | 20 1072 65792 4197376 268451840 17180065792

Peter Kagey, Illustration of T(3,2)=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-1, A-3.

Cf. A225910, A367524, A368219, A368220, A368221.

A368218[n_, m_] := 2^(n*m/2 - 2)*(2^(n*m/2) + Boole[EvenQ[n*m]] + Boole[EvenQ[m]] + If[EvenQ[n], 1, 2^(m/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).

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.

A367535 The number of ways of tiling the n X n torus 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, 16, 3692, 33570410, 5629501212064, 16397105856182791856, 808450637900676611412052288, 664613997892457939442293683754387488, 9021615045252487149405529092893182593313188608, 2008672555323737844427452615613431716686417747867226446336

Offset: 1

Peter Kagey, Dec 13 2023

nonn

Peter Kagey, Illustration of a(2)=16
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-22.

Cf. A255016, A295223, A367524, A367533, A367534.

A367535[n_] := 1/(8 n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] +If[OddQ[n], 0, n^2 (3*2^(n^2 - 2) + 2^(n^2/2)) ] +2*If[EvenQ[n], n/2*DivisorSum[n, Function[c, EulerPhi[c] (4^(n*n/LCM[2, c]) + 4^((n - 2)*n/LCM[2, c]) If[OddQ[c], 2, 4]^(2 n/c))]], n*DivisorSum[n, Function[c, EulerPhi[c] (4^((n - 1)*n/LCM[2, c]) If[OddQ[c], 2, 4]^(n/c))]]] +n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 0, EvenQ[d], 2^(n^2/d + 1)]]])

A368137 Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under 180-degree rotation.

Original entry on oeis.org

1, 23, 3776, 33601130, 5629507922944, 16397105889110874288, 808450637900797243544928256, 664613997892457948377435344457451552, 9021615045252487149406066393257455761827823616, 2008672555323737844427452616231411384297679581096869206528

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, p. A-23.

Cf. A255016, A295223, A367524, A367533, A367534, A367535, A367536, A368138.

A368137[n_] := 1/(8 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], 19*2^(n^2 - 2) + 2^(n^2/2), 2^(n^2 + 1)] + n*If[EvenQ[n], DivisorSum[n, Function[d, EulerPhi[ d] (If[EvenQ[d], 2 (2^(n^2/d) + 4^(n^2/d)), 2^(n^2/d)])]], DivisorSum[n, Function[d, EulerPhi[d] (If[EvenQ[d], 2 (2^(n^2/d) + 4^(n^2/d)), 0])]]])

cp's OEIS Frontend

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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A368218 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under horizontal reflection only.

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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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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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A367535 The number of ways of tiling the n X n torus 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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A368137 Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under 180-degree rotation.

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