OFFSET
1,20
COMMENTS
Two of 17 different symmetry types comprising A007173 and A027610. Type N is one of 10 for A371351; type O one of 7 for A371350. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type N achiral symmetry or type O chiral symmetry and n tetrahedral cells. The axis of threefold rotational symmetry is the altitude of a tetrahedron (32); the order of the symmetry group is 6. For type N, the two rooted polyominoes sharing the central face are a chiral pair reflected in that face; for type O they have the same orientation. - Robert A. Russell, Mar 22 2024
LINKS
L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047766 examples
FORMULA
G.f.: (z^2*G(z^6) - z^2*G(z^12) - z^8*G(z^12)^2) / 2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024
MATHEMATICA
Table[Switch[Mod[n, 12], 2, 3Binomial[(n-2)/2, (n-2)/6]/(2n+2)-3Binomial[(n-2)/4, (n-2)/12]/(n+4), 8, 3Binomial[(n-2)/2, (n-2)/6]/(2n+2)-6Binomial[(n-4)/4, (n-2)/6]/(n+4), _, 0], {n, 50}] (* Robert A. Russell, Mar 22 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
2nd A-number in the formula corrected by R. J. Mathar, Oct 21 2008
STATUS
approved