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.

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

Offset: 1

Author

Peter Kagey, Dec 10 2023

Keywords

• nonn

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

Formula

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).