cp's OEIS Frontend

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

%I A385149 #14 Aug 09 2025 09:56:45
%S A385149 0,0,0,0,1,8,43,225,1162,6081,32315,174856,961764,5369567,30373643,
%T A385149 173811011,1004802212,5861460314,34468644574,204161097084,
%U A385149 1217143092549,7299002607829,44005589820244,266608357403244,1622502342468552,9914884364399700
%N A385149 Number of chiral pairs of asymmetric polyominoes with n cells of the regular tiling with Schläfli symbol {4,oo}.
%C A385149 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.
%H A385149 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H A385149 Robert A. Russell, <a href="/A385149/a385149_1.pdf">Stereographic projections of chiral pairs of asymmetric polyominoes with 4 or 5 cells</a>
%F A385149 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.
%e A385149  __ __ __    __ __ __
%e A385149 |__|__|__|  |__|__|__|  a(4) = 1.
%e A385149       |__|  |__|
%t A385149 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}]
%Y A385149 Cf. A005034 (oriented), A005036 (unoriented), A369315 (chiral), A047749 (achiral), A001764 (rooted).
%K A385149 nonn,easy
%O A385149 0,6
%A A385149 _Robert A. Russell_, Jun 19 2025