A047753 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type I.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 7, 0, 0, 0, 12, 0, 0, 0, 29, 0, 0, 0, 55, 0, 0, 0, 143, 0, 0, 0, 271, 0, 0, 0, 728, 0, 0, 0, 1428, 0, 0, 0, 3873, 0, 0, 0, 7752, 0, 0, 0, 21318, 0, 0, 0, 43256, 0, 0, 0, 120175, 0, 0, 0, 246675, 0, 0, 0

Offset

1,13

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 I achiral symmetry and n tetrahedral cells. The center of symmetry is the center of a tetrahedral cell (3.1); the order of the symmetry group is 8. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

Links

Table of n, a(n) for n=1..76.

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 A047753 example.

Formula

If n=4m+1 then A047749(m) - A047751(n), otherwise 0.

G.f.: z*G(z^8) + z^5*G(z^8)^2 - 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[Switch[Mod[n, 8], 1, 4Binomial[(3n-3)/8, (n-1)/8]/(n+3)-If[17==Mod[n, 24], 24Binomial[(n-9)/8, (n-17)/24]/(n+7), 0], 5, 4Binomial[(3n-7)/8, (n+3)/8]/(n-1)-If[5==Mod[n, 24], 12Binomial[(n-5)/8, (n-5)/12]/(n+7), 0], _, 0]-Boole[1==n], {n, 50}] (* Robert A. Russell, Mar 22 2024 *)

Crossrefs

Cf. A047767.

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047751 (type K), A047749 (type U).

Sequence in context: A269243 A036274 A359241 * A303947 A031123 A359542

Adjacent sequences: A047750 A047751 A047752 * A047754 A047755 A047756

Keyword

nonn

Author

N. J. A. Sloane

Status

approved