%I #7 Mar 21 2024 08:32:10

%S 0,0,0,1,3,16,78,397,2037,10820,58349,320824,1790189,10125858,

%T 57938771,334941363,1953830203,11489589280,68053757016,405714603234,

%U 2433001205088,14668531344984,88869454457853,540834122500464

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

%C 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 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 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 a(n) = A007173(n) - A027610(n) = (A007173(n) - A371351(n))/2 = A027610(n) - A371351(n).

%F a(n) = h(3,n) - H(3,n) in Table 8 of Hering link.

%F 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 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 Sum of chiral symmetry types (A047776, A047774, A047762, A047758, A047752, A047769, A047766 [type O]) in Beineke article.

%Y Cf. A007173 (oriented), A027610 (oriented), A371351 (achiral), A001764 (rooted), A369314 {3,oo}, A369474 {3,3,3,oo}.

%K nonn

%O 1,5

%A _Robert A. Russell_, Mar 19 2024