A371351 Number of achiral polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}.

1, 1, 1, 2, 4, 8, 15, 37, 73, 182, 364, 952, 1944, 5169, 10659, 28842, 60115, 164450, 345345, 953814, 2016144, 5609760, 11920740, 33378072, 71250060, 200553733, 429757960, 1215177680, 2612635888, 7416503776

Offset: 1

Author

Robert A. Russell, Mar 19 2024

Keywords

• nonn

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.

Crossrefs

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

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

Programs

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}]

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*A027610(n) - A007173(n) = A007173(n) - 2*A371350(n) = A027610(n) - A371350(n).

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.