A369314 - OEIS

%I #7 Jan 22 2024 00:04:03

%S 1,2,7,22,68,214,691,2240,7396,24702,83469,284928,981814,3410990,

%T 11939752,42075308,149180356,531866972,1905872189,6861162880,

%U 24805796984,90035940942,327988261992,1198853954688,4395798528850

%N Number of chiral pairs of polyominoes composed of n triangular cells of the hyperbolic regular tiling with Schläfli symbol {3,oo}.

%C A stereographic projection of the {3,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) = C(2n,2)/(2(n+1)(n+2)) - [2\(n+1)]*C(n,(n+1)/2)/(2n) - [2\n]*C(n,n/2)/(2n+4) + [3\(n-1)]*C((2n+1)/3,(n-1)/3)/(2n+1).

%F a(n) = A001683(n+2) - A000207(n) = (A001683(n+2) - A208355(n-1)) / 2 = A000207(n) - A208355(n-1).

%e ________      ________   ________      ________   ________      ________

%e \  /\  /\    /\  /\  /   \  /\  /\    /\  /\  /   \  /\  /\    /\  /\  /

%e  \/__\/__\  /__\/__\/     \/__\/__\  /__\/__\/     \/__\/__\  /__\/__\/

%e                            \  /          \  /           \  /  \  /

%e a(4)=1; a(5)=2.             \/            \/             \/    \/

%t Table[Binomial[2n,n]/(2(n+1)(n+2))-If[OddQ[n],Binomial[n,(n+1)/2]/n,Binomial[n,n/2]/(n+2)]/2+If[Divisible[n-1,3],Binomial[(2n+1)/3,(n-1)/3]/(2n+1),0],{n,4,20}]

%Y Polyominoes: A001683(n+2) (oriented), A000207 (unoriented), A208355(n-1) (achiral).

%K nonn,easy

%O 4,2

%A _Robert A. Russell_, Jan 19 2024