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

%I A369315 #12 Jun 20 2025 10:46:26
%S A369315 2,9,48,231,1188,6114,32448,175032,962472,5370524,30377504,173816313,
%T A369315 1004823816,5861490300,34468767840,204161269620,1217143807770,
%U A369315 7299003615537,44005594027200,266608363362900
%N A369315 Number of chiral pairs of polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}.
%C A369315 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 A369315 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%F A369315 a(n) = A005034(n) - A005036(n) = (A005034(n) - A047749(n)) / 2 = A005036(n) - A047749(n).
%F A369315 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
%e A369315  __ __ __    __ __ __     __ __          __ __
%e A369315 |__|__|__|  |__|__|__|   |__|__|__    __|__|__|  a(4) = 2.
%e A369315       |__|  |__|            |__|__|  |__|__|
%t A369315 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}]
%t A369315 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 *)
%Y A369315 Polyominoes: A005034 (oriented), A005036 (unoriented), A047749 (achiral), A385149 (asymmetric), A001764 (rooted), A369314 {3,oo}.
%K A369315 easy,nonn
%O A369315 4,1
%A A369315 _Robert A. Russell_, Jan 19 2024