A047769 - OEIS

%I #19 Apr 01 2024 17:24:05

%S 0,0,0,1,0,5,0,26,0,133,0,708,0,3860,0,21604,0,123266,0,715216,0,

%T 4206956,0,25032840,0,150413322,0,911379384,0,5562367173,0,

%U 34164355715,0,211015212580,0,1309815397995,0,8166460799097,0,51120054233490

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

%C 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

%H L. W. Beineke and R. E. Pippert, <a href="http://dx.doi.org/10.4153/CJM-1974-006-x">Enumerating dissectable polyhedra by their automorphism groups</a>, Canad. J. Math., 26 (1974), 50-67.

%H Robert A. Russell, <a href="/A047769/a047769.txt">Mathematica Graphics3D program for A047769 example</a>

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

%F 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

%t 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,

%t (n-8)/12]]/(n+4),0])/2],{n,60}] (* _Robert A. Russell_, Mar 22 2024 *)

%Y Cf. A047770.

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

%K nonn

%O 1,6

%A _N. J. A. Sloane_