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

1,2

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.

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).

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 *)

Crossrefs

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

Sequence in context: A012076 A173211 A109901 * A015018 A204245 A287248

Adjacent sequences: A367519 A367520 A367521 * A367523 A367524 A367525

Keyword

nonn

Author

Peter Kagey, Nov 21 2023

Status

approved