A047760 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type F.

0, 0, 1, 0, 1, 0, 3, 0, 5, 0, 12, 0, 23, 0, 55, 0, 114, 0, 273, 0, 588, 0, 1428, 0, 3156, 0, 7752, 0, 17427, 0, 43263, 0, 98516, 0, 246675, 0, 567281, 0, 1430715, 0, 3316521, 0, 8414640, 0, 19633796, 0, 50067108, 0, 117464424, 0, 300830572, 0, 709098696

Offset

1,7

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 F chiral symmetry and n tetrahedral cells. The axis of symmetry connects opposite edge centers of a tetrahedron (31); 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..53.

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 A047760 examples

Formula

If n=2m+1 then (1/2)*(A047750(m) - A047753(n) - A047751(n)), otherwise 0.

G.f.: (2*(G(z^4)-1)/z - z*G(z^4) + 2z*G(z^4)^2 - z*G(z^8) - z^5*G(z^8)^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, 4], 3, 6Binomial[(3n-1)/4, (n+1)/2]/(n+3), 1, (5n-1)Binomial[(3n-3)/4, (n-1)/4]/((n+1)(n+3))-Switch[Mod[n, 8], 1, 2Binomial[(3n-3)/8, (n-1)/8]/(n+3), 5, 2Binomial[(3n-7)/8, (n+3)/8]/(n-1), _, 0], _, 0], {n, 60}] (* Robert A. Russell, Mar 22 2024 *)

Crossrefs

Cf. A047761, A047750.

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047751 (type K), A047753 (type I).

Sequence in context: A049689 A227901 A118657 * A276908 A242246 A229979

Adjacent sequences: A047757 A047758 A047759 * A047761 A047762 A047763

Keyword

nonn

Author

N. J. A. Sloane

Status

approved