A295229 Number of tilings of the n X n grid, using diagonal lines to connect the grid points.

1, 6, 84, 8548, 4203520, 8590557312, 70368815480832, 2305843028004192256, 302231454912728264605696, 158456325028538104598816096256, 332306998946228986960926214931349504, 2787593149816327892769293535238052808491008

Offset: 1

Author

Peter Kagey, Nov 18 2017

Keywords

• nonn

Comments

The grids are counted up to reflection and rotation.

a(n) <= A295223(n).

Examples

			For n = 2, the a(2) = 6 tilings are:
//, \/, /\, \\, /\, and \/.
//  //  //  //  \/      /\
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..57
• Andrew Howroyd, Derivation of Formula
• Peter Kagey, Example of the 6 tilings of the 2 X 2 grid.
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.

Crossrefs

Cf. A054247, A295223.

Programs

Mathematica

Array[(2^(#^2) + 2*2^(# (# + 1)/2) + If[EvenQ@ #, 3*2^(#^2/2) + 2*2^(#^2/4), 2^((#^2 + 1)/2)])/8 &, 12] (* Michael De Vlieger, Apr 12 2018 *)

PARI

a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + if(n%2, 2^((n^2+1)/2), 3*2^(n^2/2) + 2*2^(n^2/4)))/8; \\ Andrew Howroyd, Nov 19 2017

Formula

From Andrew Howroyd, Nov 19 2017: (Start)

a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + 3*2^(n^2/2) + 2*2^(n^2/4)) / 8 for n even.

a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + 2^((n^2+1)/2)) / 8 for n odd. (End)

Extensions

a(5)-a(12) from Andrew Howroyd, Nov 19 2017