A369314 Number of chiral pairs of polyominoes composed of n triangular cells of the hyperbolic regular tiling with Schläfli symbol {3,oo}.

1, 2, 7, 22, 68, 214, 691, 2240, 7396, 24702, 83469, 284928, 981814, 3410990, 11939752, 42075308, 149180356, 531866972, 1905872189, 6861162880, 24805796984, 90035940942, 327988261992, 1198853954688, 4395798528850

Offset: 4

Author

Robert A. Russell, Jan 19 2024

Keywords

• nonn
• easy

Comments

A stereographic projection of the {3,oo} tiling on the Poincaré disk can be obtained via the Christensson link. Each member of a chiral pair is a reflection but not a rotation of the other.

Examples

			________      ________   ________      ________   ________      ________
\  /\  /\    /\  /\  /   \  /\  /\    /\  /\  /   \  /\  /\    /\  /\  /
 \/__\/__\  /__\/__\/     \/__\/__\  /__\/__\/     \/__\/__\  /__\/__\/
                           \  /          \  /           \  /  \  /
a(4)=1; a(5)=2.             \/            \/             \/    \/
		

Links

• Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Crossrefs

Polyominoes: A001683(n+2) (oriented), A000207 (unoriented), A208355(n-1) (achiral).

Programs

Mathematica

Table[Binomial[2n,n]/(2(n+1)(n+2))-If[OddQ[n],Binomial[n,(n+1)/2]/n,Binomial[n,n/2]/(n+2)]/2+If[Divisible[n-1,3],Binomial[(2n+1)/3,(n-1)/3]/(2n+1),0],{n,4,20}]

Formula

a(n) = C(2n,2)/(2(n+1)(n+2)) - [2\(n+1)]*C(n,(n+1)/2)/(2n) - [2\n]*C(n,n/2)/(2n+4) + [3\(n-1)]*C((2n+1)/3,(n-1)/3)/(2n+1).

a(n) = A001683(n+2) - A000207(n) = (A001683(n+2) - A208355(n-1)) / 2 = A000207(n) - A208355(n-1).