A047766 Number of dissectable polyhedra with n tetrahedral cells with symmetry of type N or chiral pairs with symmetry of type O.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 26, 0, 0, 0, 0, 0, 133, 0, 0, 0, 0, 0, 708, 0, 0, 0, 0, 0, 3861, 0, 0, 0, 0, 0, 21604, 0, 0, 0, 0, 0, 123266, 0, 0, 0, 0, 0, 715221, 0, 0, 0, 0, 0, 4206956, 0, 0, 0, 0, 0, 25032840, 0, 0, 0, 0

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

Table of n, a(n) for n=1..78.

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

If n=6m+2 then (1/2)*(A001764(m) - A047764(n)), otherwise 0.

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

Cf. A047768.

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A371351 (achiral), A001764 (rooted), A047764 (type Q).

Sequence in context: A088194 A340978 A048894 * A336881 A369449 A324323

Adjacent sequences: A047763 A047764 A047765 * A047767 A047768 A047769

Keyword

nonn,changed

Author

N. J. A. Sloane

Extensions

2nd A-number in the formula corrected by R. J. Mathar, Oct 21 2008

Status

approved