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Number of achiral polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}.
14

%I #6 Mar 21 2024 08:32:14

%S 1,1,1,2,4,8,15,37,73,182,364,952,1944,5169,10659,28842,60115,164450,

%T 345345,953814,2016144,5609760,11920740,33378072,71250060,200553733,

%U 429757960,1215177680,2612635888,7416503776

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

%C Also number of achiral simplicial 3-clusters or stack polytopes with n tetrahedral cells. An achiral polyomino is identical to its reflection.

%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) = ([0==n mod 2]*2*C(3n/2,n) + [1==n mod 2]*3*C((3n-1)/2,n) + [1==n mod4]*3*C((3n-3)/4,(n-1)/2) + [2==n mod6]*3*C(n/2-1,(n-2)/3)) / (3n+3).

%F a(n) = 2*A027610(n) - A007173(n) = A007173(n) - 2*A371350(n) = A027610(n) - A371350(n).

%F a(n) = 2*H(3,n) - h(3,n) in Table 8 of Hering link.

%F G.f.: (-4 + 4*G(z^2) + 3z*G(z^2)^2 + 3z*G(z^4) + 2z^2*G(z^6)) / 6, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764.

%t Table[(If[OddQ[n],3Binomial[(3n-1)/2,n],2Binomial[3n/2,n]]+If[1==Mod[n,4],3Binomial[(3n-3)/4,(n-1)/2],0]+If[2==Mod[n,6],3Binomial[n/2-1,(n-2)/3],0])/(3n+3),{n,30}]

%Y Sum of achiral symmetry types (A047775, A047773, A047760, A047754, A047753, A047751, A047771, A047766 [type N], A047765, A047764) in Beineke link.

%Y Cf. A007173 (oriented), A027610 (oriented), A371350 (chiral), A001764 (rooted), A208355(n-1) {3,oo}, A182299 {3,3,3,oo}.

%K nonn

%O 1,4

%A _Robert A. Russell_, Mar 19 2024