A047751 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type K.

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0

Offset

1,41

Comments

One of 17 different symmetry types comprising A007173 and A027610 and one of 10 for A371351. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type K achiral symmetry and n tetrahedral cells. The center of symmetry is the center of a tetrahedral cell (3); the order of the symmetry group is 24. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

Links

Table of n, a(n) for n=1..91.

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 A047751 examples

Formula

a(1)=1, a(n)=0 unless n == 5 (mod 12); a(12m+5) = A047749(m).

G.f.: z + z^5*G(z^24) + z^17*G(z^24)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024

Mathematica

Table[Boole[1==n]+Switch[Mod[n, 24], 5, 12Binomial[(n-5)/8, (n-5)/12], 17, 24Binomial[(n-9)/8, (n-17)/24], _, 0]/(n+7), {n, 60}] (* Robert A. Russell, Mar 22 2024 *)

Crossrefs

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted).

Sequence in context: A028637 A070208 A028629 * A028706 A060175 A358480

Adjacent sequences: A047748 A047749 A047750 * A047752 A047753 A047754

Keyword

nonn

Author

N. J. A. Sloane

Status

approved