OFFSET
1,4
COMMENTS
Also number of achiral simplicial 3-clusters or stack polytopes with n tetrahedral cells. An achiral polyomino is identical to its reflection.
LINKS
L. W. Beineke and R. E. Pippert Enumerating dissectable polyhedra by their automorphism groups, Can. J. Math., 26 (1974), 50-67
F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.
FORMULA
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).
a(n) = 2*H(3,n) - h(3,n) in Table 8 of Hering link.
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.
MATHEMATICA
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert A. Russell, Mar 19 2024
STATUS
approved