A047752 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type J.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 133, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 708, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3861, 0

Offset

1,41

Comments

One of 17 different symmetry types comprising A007173 and A027610 and 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 J chiral symmetry and n tetrahedral cells. The center of symmetry is the center of a tetrahedral cell (3); the order of the symmetry group is 12. Each member of a chiral pair is a reflection but not a rotation of the other. - Robert A. Russell, Mar 22 2024

Links

Table of n, a(n) for n=1..90.

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 A047752 example

Formula

If n=12m+5 then (1/2)*(A001764(m) - A047751(n)), otherwise 0.

G.f.: z^5 * (G(z^12) - G(z^24) - z^12*G(z^24)^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, 24], 5, 6Binomial[(n-5)/4, (n-5)/12]/(n+1)-12Binomial[(n-5)/8, (n-5)/12]/(n+7), 17, 6Binomial[(n-5)/4, (n-5)/12]/(n+1)-24Binomial[(n-9)/8, (n-17)/24]/(n+7), _, 0]/2, {n, 60}] (* Robert A. Russell, Mar 22 2024 *)

Crossrefs

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047751 (type K).

Sequence in context: A341881 A281462 A236239 * A088194 A340978 A048894

Adjacent sequences: A047749 A047750 A047751 * A047753 A047754 A047755

Keyword

nonn

Author

N. J. A. Sloane

Status

approved