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A047764 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type Q.

%I A047764 #21 Mar 31 2024 14:03:23
%S A047764 0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,7,0,0,
%T A047764 0,0,0,12,0,0,0,0,0,30,0,0,0,0,0,55,0,0,0,0,0,143,0,0,0,0,0,273,0,0,0,
%U A047764 0,0,728,0,0,0,0,0,1428,0,0,0,0,0,3876,0,0,0,0,0,7752
%N A047764 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type Q.
%C A047764 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 Q achiral symmetry and n tetrahedral cells. The center of symmetry is the center of a tetrahedral face (2.10); the order of the symmetry group is 12. An achiral polyomino is identical to its reflection. - _Robert A. Russell_, Mar 22 2024
%H A047764 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 A047764 Robert A. Russell, <a href="/A047764/a047764.txt">Mathematica Graphics3D program for A047764 examples</a>
%F A047764 If n=6m+2 then A047749((n-2)/6), otherwise 0.
%F A047764 G.f.: z^2*G(z^12) + z^8*G(z^12)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - _Robert A. Russell_, Mar 22 2024
%t A047764 Table[Switch[Mod[n,12],2,6Binomial[(n-2)/4,(n-2)/12],8,12Binomial[(n-4)/4,(n-2)/6],_,0]/(n+4),{n,50}] (* _Robert A. Russell_, Mar 22 2024 *)
%Y A047764 Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U).
%K A047764 nonn
%O A047764 1,20
%A A047764 _N. J. A. Sloane_