A047769 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type M.

0, 0, 0, 1, 0, 5, 0, 26, 0, 133, 0, 708, 0, 3860, 0, 21604, 0, 123266, 0, 715216, 0, 4206956, 0, 25032840, 0, 150413322, 0, 911379384, 0, 5562367173, 0, 34164355715, 0, 211015212580, 0, 1309815397995, 0, 8166460799097, 0, 51120054233490

Offset

1,6

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 M chiral symmetry and n tetrahedral cells. The axis of symmetry is the altitude of a tetrahedral face (21); the order of the symmetry group is 2. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

Links

Table of n, a(n) for n=1..40.

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 A047769 example

Formula

If n=2m then (1/2)*(A001764(m) - 2*A047766(n) - A047765(n) - A047764(n)), otherwise 0.

G.f.: (G(z^2) - G(z^4) - z^2 * (G(z^4)^2 + G(z^6) - G(z^12) - z^6*G(z^12)^2)) / 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, (Binomial[3n/2, n/2]/(n+1)-If[0==Mod[n, 4], 2Binomial[3n/4, n/4]/(n+2), 4Binomial[(3n-2)/4, (n-2)/4]/(n+2), 0]- If[2==Mod[n, 6], 3Binomial[3(n-2)/6, (n-2)/6]/(n+1)- If[2==Mod[n, 12], 6Binomial[3(n-2)/12, (n-2)/12], 12Binomial[(3n-12)/12,
(n-8)/12]]/(n+4), 0])/2], {n, 60}] (* Robert A. Russell, Mar 22 2024 *)

Crossrefs

Cf. A047770.

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047764 (type Q), A047765 (type P), A047766 (type O).

Sequence in context: A167315 A167362 A070611 * A279501 A279548 A279701

Adjacent sequences: A047766 A047767 A047768 * A047770 A047771 A047772

Keyword

nonn

Author

N. J. A. Sloane

Status

approved