A047765 - OEIS

A047765

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

0, 0, 0, 1, 0, 2, 0, 2, 0, 7, 0, 12, 0, 29, 0, 55, 0, 143, 0, 271, 0, 728, 0, 1428, 0, 3873, 0, 7752, 0, 21318, 0, 43256, 0, 120175, 0, 246675, 0, 690678, 0, 1430715, 0, 4032015, 0, 8414610, 0, 23841480, 0, 50067108, 0, 142498637, 0, 300830572

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

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 P achiral symmetry and n tetrahedral cells. The center of symmetry is the altitude of a tetrahedral face (21); the order of the symmetry group is 4. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

LINKS

Table of n, a(n) for n=1..52.

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 A047765 examples

FORMULA

If n=2m then A047749(m) - A047764(n), otherwise 0.

G.f.: G(z^4) + z^2*G(z^4)^2 - z^2*G(z^12) - z^8*G(z^12)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024

MATHEMATICA

Table[If[OddQ[n], 0, If[OddQ[n/2], 2Binomial[(3n-2)/4, (n-2)/4], Binomial[3n/4, n/4]]/(n/2+1)-Switch[Mod[n, 12], 2, 6Binomial[(n-2)/4, (n-2)/12], 8, 12Binomial[(n-4)/4, (n-2)/6], _, 0]/(n+4)], {n, 52}] (* Robert A. Russell, Mar 22 2024 *)

CROSSREFS

Cf. A047767.

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U), A047764 (type Q).

Sequence in context: A066285 A327873 A136665 * A068463 A099554 A319697

Adjacent sequences: A047762 A047763 A047764 * A047766 A047767 A047768

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved