cp's OEIS Frontend

A371350 Number of chiral pairs of polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}.

%I A371350 #11 Dec 05 2024 19:12:10
%S A371350 0,0,0,1,3,16,78,397,2037,10820,58349,320824,1790189,10125858,
%T A371350 57938771,334941363,1953830203,11489589280,68053757016,405714603234,
%U A371350 2433001205088,14668531344984,88869454457853,540834122500464
%N A371350 Number of chiral pairs of polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}.
%C A371350 Also number of chiral pairs of simplicial 3-clusters or stack polytopes with n tetrahedral cells. Each member of a chiral pair is a reflection but not a rotation of the other.
%H A371350 L. W. Beineke and R. E. Pippert <a href="http://dx.doi.org/10.4153/CJM-1974-006-x">Enumerating dissectable polyhedra by their automorphism groups</a>, Can. J. Math., 26 (1974), 50-67
%H A371350 F. Hering et al., <a href="http://dx.doi.org/10.1016/0012-365X(82)90121-2">The enumeration of stack polytopes and simplicial clusters</a>, Discrete Math., 40 (1982), 203-217.
%F A371350 a(n) = A007173(n) - A027610(n) = (A007173(n) - A371351(n))/2 = A027610(n) - A371351(n).
%F A371350 a(n) = h(3,n) - H(3,n) in Table 8 of Hering link.
%F A371350 G.f.: (4*G(z) - 2*G(z)^2 + z*G(z)^4 - 2*G(z^2) - 3z*G(z^2)^2 + 2z*(4 G(z^3) + 2z*G(z^3)^2 - 3*G(z^4) - 2z*G(z^6))) / 24.
%t A371350 Table[Switch[Mod[n,3],1,Binomial[n,(n-1)/3],2,Binomial[n,(n-2)/3],_,0]/(3n)+(Binomial[3n,n]/(6n+3)-If[OddQ[n],Binomial[3(n-1)/2+1,n],Binomial[3n/2,n]/3]-2If[1==Mod[n,4],Binomial[(3n-3)/4,(n-1)/2],0]-2If[2==Mod[n,6],Binomial[n/2-1,n/3-2/3],0])/(4n+4),{n,30}]
%Y A371350 Sum of chiral symmetry types (A047776, A047774, A047762, A047758, A047752, A047769, A047766 [type O]) in Beineke article.
%Y A371350 Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A369314 {3,oo}, A369474 {3,3,3,oo}.
%K A371350 nonn
%O A371350 1,5
%A A371350 _Robert A. Russell_, Mar 19 2024