A047762 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type E.

0, 0, 0, 0, 1, 0, 6, 0, 32, 0, 176, 0, 952, 0, 5302, 0, 29960, 0, 172536, 0, 1007575, 0, 5959656, 0, 35622384, 0, 214875104, 0, 1306303424, 0, 7995896502, 0, 49236826080, 0, 304799714960, 0, 1895785216039, 0, 11841367945110, 0, 74245791718824

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

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 E 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 2. Each member of a chiral pair is a reflection but not a rotation of the other. - Robert A. Russell, Mar 22 2024

Links

• 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 A047762 example.

Crossrefs

Cf. A047749, A047763.

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047751 (type K), A047752 (type J), A047753 (type I), A047754 (type H), A047758 (type G), A047760 (type F).

Programs

Mathematica

Table[If[OddQ[n],Binomial[(3n-1)/2,(n-1)/2]/(n+1)-If[1==Mod[n,4],Binomial[(3n-3)/4,(n-1)/4]/((n+1))+(4Binomial[(3n+1)/4,(n-1)/4]-If[1==Mod[n,8],4Binomial[(3n-3)/8,(n-1)/8],8Binomial[(3n-7)/8,(n-5)/8]])/(n+3),2Binomial[(3n+3)/4,(n+1)/4]/(n+3)],0]/2,{n,40}] (* Robert A. Russell, Mar 22 2024 *)

Formula

If n=2m+1 then (1/4)*(A047749(n) - 2*A047760(n) - 6*A047758(n) - 2*A047754(n) - 3*A047753(n) - 2*A047752(n) - A047751(n)), otherwise 0.

G.f.: (z^2*G(z^2)^2 - (2+z^2)*G(z^4) - 2*z^2*G(z^4)^2 + 2*(1 + z^2*G(z^8) + z^6*G(z^8)^2)) / (4*z), where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024