A047937 Number of 2-colorings of an n X n grid, up to rotational symmetry.

1, 2, 6, 140, 16456, 8390720, 17179934976, 140737496748032, 4611686019501162496, 604462909807864344215552, 316912650057057631849169289216, 664613997892457937028364283517337600, 5575186299632655785385110159782842147536896, 187072209578355573530071668259090783437390809661440

Offset

0,2

Comments

Cycle index = 1/4(s_1^(n^2)+ 2 s_4^floor(n^2/4)s_1^(n mod 2)+s_2^floor(n^2/2)s_1^(n mod 2)). - Geoffrey Critzer, Oct 28 2011

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Peter Kagey, Illustration of a(3)=140

Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.

Formula

a(n) = (m^(n^2) + 2*m^((n^2 + 3*(n mod 2))/4) + m^((n^2 + (n mod 2))/2))/4, with m = 2.

Example

a(2)=6 from
00 10 11 10 11 11
00 00 00 01 10 11

Mathematica

Table[(2^(n^2)+2*2^Floor[n^2/4]*2^Mod[n, 2]+2^Floor[n^2/2]*2^Mod[n, 2])/4, {n, 0, 10}]  (* Geoffrey Critzer, Oct 28 2011 *)

Crossrefs

Column k=2 of A343095.

Other columns are A047938, A047939, A047940, A047941, A047942, A047943, A047944, A047945.

Cf. A054247.

Sequence in context: A368395 A090907 A159478 * A027731 A280821 A145143

Adjacent sequences: A047934 A047935 A047936 * A047938 A047939 A047940

Keyword

nonn,easy,nice

Author

Rob Pratt

Extensions

Terms a(12) and beyond from Andrew Howroyd, Apr 14 2021

Status

approved