A385149 Number of chiral pairs of asymmetric polyominoes with n cells of the regular tiling with Schläfli symbol {4,oo}.

0, 0, 0, 0, 1, 8, 43, 225, 1162, 6081, 32315, 174856, 961764, 5369567, 30373643, 173811011, 1004802212, 5861460314, 34468644574, 204161097084, 1217143092549, 7299002607829, 44005589820244, 266608357403244, 1622502342468552, 9914884364399700

Offset: 0

Author

Robert A. Russell, Jun 19 2025

Keywords

• nonn
• easy

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) = 1.
      |__|  |__|
		

Links

• Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
• Robert A. Russell, Stereographic projections of chiral pairs of asymmetric polyominoes with 4 or 5 cells

Crossrefs

Cf. A005034 (oriented), A005036 (unoriented), A369315 (chiral), A047749 (achiral), A001764 (rooted).

Programs

Mathematica

Table[If[n<4,0,(3Binomial[3n,n]/(2n+1)-Binomial[3n+1,n]/(n+1) + Switch[Mod[n,4], 0,4Binomial[3n/4,n/4]/(n/2+1)-6Binomial[3n/2,n/2]/(n+1), 1,(4Binomial[(3n-3)/4,(n-1)/4]-10Binomial[(3n-1)/2,(n-1)/2])/(n+1)+(8Binomial[(3n+1)/4,(n-1)/4]+16Binomial[(3n-3)/4,(n-5)/4])/(n+3), 2,16Binomial[(3n-2)/4,(n-2)/4]/(n+2)-6Binomial[3n/2,n/2]/(n+1), 3,24Binomial[(3n-1)/4,(n-3)/4]/(n+3)-10Binomial[(3n-1)/2,(n-1)/2]/(n+1)])/8],{n,0,30}]

Formula

G.f.: (3*G(z) - G(z)^2 - 6*G(z^2) - 5z*G(z^2)^2 + 4*G(z^4) + 2z*G(z^4) + 2z*G(z^4)^2 + 4z^2*G(z^4)^2 + 4z^3*G(z^4)^3 + 2z^5*G(z^4)^4) / 8, where G(z)=1+z*G(z)^3 is the g.f. for A001764.