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%I #17 Mar 22 2024 17:03:33
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,26,0,0,0,0,0,133,0,
%T 0,0,0,0,708,0,0,0,0,0,3861,0,0,0,0,0,21604,0,0,0,0,0,123266,0,0,0,0,
%U 0,715221,0,0,0,0,0,4206956,0,0,0,0,0,25032840,0,0,0,0
%N Number of dissectable polyhedra with n tetrahedral cells with symmetry of type N or chiral pairs with symmetry of type O.
%C Two of 17 different symmetry types comprising A007173 and A027610. Type N is one of 10 for A371351; type O 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 N achiral symmetry or type O chiral 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 6. For type N, the two identical rooted polyominoes sharing the central face are a chiral pair reflected in that face; for type O they have the same orientation. - _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.
%F If n=6m+2 then (1/2)*(A001764(m) - A047764(n)), otherwise 0.
%F G.f.: (z^2*G(z^6) - z^2*G(z^12) - z^8*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[Switch[Mod[n,12],2,3Binomial[(n-2)/2,(n-2)/6]/(2n+2)-3Binomial[(n-2)/4,(n-2)/12]/(n+4),8,3Binomial[(n-2)/2,(n-2)/6]/(2n+2)-6Binomial[(n-4)/4,(n-2)/6]/(n+4),_,0],{n,50}] (* _Robert A. Russell_, Mar 22 2024 *)
%Y Cf. A047768.
%Y Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A371351 (achiral), A001764 (rooted), A047764 (type Q).
%K nonn
%O 1,20
%A _N. J. A. Sloane_
%E 2nd A-number in the formula corrected by _R. J. Mathar_, Oct 21 2008
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