A302484 -id:A302484 - cp's OEIS frontend

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.

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).

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).

A367536 Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under matrix transposition but no other symmetries.

Original entry on oeis.org

1, 17, 3692, 33572458, 5629501212064, 16397105857614447792, 808450637900676611412052288, 664613997892457939730524059906099232, 9021615045252487149405529092893182593313188608, 2008672555323737844427452615629277349189270615385935288832

Offset: 1

Peter Kagey, Dec 13 2023

nonn

A Truchet tile is an example of a tile that is fixed under matrix transposition but no other symmetries.

Peter Kagey, Illustration of a(2)=17
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-23.
Eric Weisstein's World of Mathematics, Truchet Tiling

Cf. A255016, A295223, A302484, A367533, A367534, A367535.

A367536[n_] := 1/(8n^2) (DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] +If[OddQ[n],n*DivisorSum[n, Function[c, EulerPhi[c] 2^(n^2/c + 1)]],n*DivisorSum[n, Function[c, EulerPhi[c] (4^(n^2/LCM[2, c]) + 2^(n^2/c + 1) + If[OddQ[c], 0, 4^(n^2/c)])]] + n^2 (3*2^(n^2 - 2) + 2^(n^2/2))])

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

A367536 Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under matrix transposition but no other symmetries.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica