cp's OEIS Frontend

A369472 Number of achiral polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.

%I A369472 #26 Jul 19 2024 08:46:39
%S A369472 1,1,2,4,9,22,52,140,340,969,2394,7084,17710,53820,135720,420732,
%T A369472 1068012,3362260,8579560,27343888,70068713,225568798,580034052,
%U A369472 1882933364,4855986044,15875338990,41043559340,134993766600
%N A369472 Number of achiral polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.
%C A369472 A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christensson link.
%H A369472 Andrew Howroyd, <a href="/A369472/b369472.txt">Table of n, a(n) for n = 1..1000</a>
%H A369472 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%F A369472 For n even, a(n) = C(2n,n/2)/(3n/2+1).
%F A369472 For n odd, a(n) = 4*C(2n-1,(n-1)/2)/(3n+1).
%F A369472 a(n+2)/a(n) ~ 256/27. a(2m+1)/a(2m) ~ 32/9; a(2m)/a(2m-1) ~ 8/3.
%F A369472 a(n) = 2*A005040(n) - A005038(n) = A005038(n) - 2*A369471(n) = A005040(n) - A369471(n).
%F A369472 G.f.: G(z^2)+z*G(z^2)^2, where G(z)=1+z*G(z)^4, the generating function for A002293.
%F A369472 a(2m) = A002293(m) ~ (4^4/3^3)^m*sqrt(4/(2*Pi*(3*m)^3)). - _Robert A. Russell_, Jul 15 2024
%t A369472 p=5; Table[If[EvenQ[n],Binomial[(p-1)n/2,n/2]/((p-2)n/2+1),If[OddQ[p],(p-1)Binomial[(p-1)n/2-1,(n-1)/2]/((p-2)n+1),p Binomial[(p-1)n/2-1/2,(n-1)/2]/((p-2)n+2)]],{n,35}]
%Y A369472 Column k=5 of A370060.
%Y A369472 Polyominoes: A005038 (oriented), A005040 (unoriented), A369471 (chiral), A002293 (rooted), A047749 {4,oo}, A143546 {6,oo}.
%K A369472 easy,nonn
%O A369472 1,3
%A A369472 _Robert A. Russell_, Jan 23 2024