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

1,7

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

Table of n, a(n) for n=1..41.

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.

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

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 *)

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).

Sequence in context: A249869 A209446 A375237 * A186977 A167316 A270607

Adjacent sequences: A047759 A047760 A047761 * A047763 A047764 A047765

Keyword

nonn

Author

N. J. A. Sloane

Status

approved