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.

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

Offset: 1

Author

Peter Kagey, Nov 21 2023

Keywords

• nonn

Comments

The a(2) = 4 tilings are

- - - - - | - |

- -, | -, - |, and | -.

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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).