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Number of chiral pairs of polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}.
5

%I #7 Jan 22 2024 00:04:32

%S 2,9,48,231,1188,6114,32448,175032,962472,5370524,30377504,173816313,

%T 1004823816,5861490300,34468767840,204161269620,1217143807770,

%U 7299003615537,44005594027200,266608363362900

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

%C 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 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.

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

%e __ __ __ __ __ __ __ __ __ __

%e |__|__|__| |__|__|__| |__|__|__ __|__|__| a(4) = 2.

%e |__| |__| |__|__| |__|__|

%t 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}]

%Y Polyominoes: A005034 (oriented), A005036 (unoriented), A047749 (achiral), A369314 {3,oo}.

%K easy,nonn

%O 4,1

%A _Robert A. Russell_, Jan 19 2024