A369315 Number of chiral pairs of polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}.

2, 9, 48, 231, 1188, 6114, 32448, 175032, 962472, 5370524, 30377504, 173816313, 1004823816, 5861490300, 34468767840, 204161269620, 1217143807770, 7299003615537, 44005594027200, 266608363362900

Offset: 4

Author

Robert A. Russell, Jan 19 2024

Keywords

• easy
• nonn

Comments

A stereographic projection of the {4,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) = 2.
      |__|  |__|            |__|__|  |__|__|
		

Links

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

Crossrefs

Polyominoes: A005034 (oriented), A005036 (unoriented), A047749 (achiral), A385149 (asymmetric), A001764 (rooted), A369314 {3,oo}.

Programs

Mathematica

p=4; Table[(Binomial[(p-1)n,n]/(((p-2)n+1)((p-2)n+2))-If[OddQ[n],If[OddQ[p],Binomial[(p-1)n/2,(n-1)/2]/n,(p+1)Binomial[((p-1)n-1)/2,(n-1)/2]/((p-2)n+2)-Binomial[((p-1)n+1)/2,(n-1)/2]/((p-1)n+1)],Binomial[(p-1)n/2,n/2]/((p-2)n+2)]+DivisorSum[GCD[p,n-1],EulerPhi[#]Binomial[((p-1)n+1)/#,(n-1)/#]/((p-1)n+1)&,#>1&])/2,{n,4,30}]
Table[(3Binomial[3n,n]/(2n+1)-Binomial[3n+1,n]/(n+1)-If[OddQ[n],6Binomial[(3n-1)/2,(n-1)/2]-If[1==Mod[n,4],4Binomial[(3n-3)/4,(n-1)/4],0],2Binomial[3n/2,n/2]]/(n+1))/8,{n,0,30}] (* Robert A. Russell, Jun 19 2025 *)

Formula

a(n) = A005034(n) - A005036(n) = (A005034(n) - A047749(n)) / 2 = A005036(n) - A047749(n).

G.f.: (3*G(z) - G(z)^2 - 2*G(z^2) - 3z*G(z^2)^2 + 2z*G(z^4)) / 8, where G(z)=1+z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Jun 19 2025