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

1,2

Comments

The grids are counted up to reflection and rotation.

a(n) <= A295223(n).

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.

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)

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A054247, A295223.

Sequence in context: A293455 A334516 A331014 * A330849 A245232 A284522

Adjacent sequences: A295226 A295227 A295228 * A295230 A295231 A295232

Keyword

nonn

Author

Peter Kagey, Nov 18 2017

Extensions

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

Status

approved