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A047760 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type F.

%I A047760 #19 Mar 31 2024 14:03:18
%S A047760 0,0,1,0,1,0,3,0,5,0,12,0,23,0,55,0,114,0,273,0,588,0,1428,0,3156,0,
%T A047760 7752,0,17427,0,43263,0,98516,0,246675,0,567281,0,1430715,0,3316521,0,
%U A047760 8414640,0,19633796,0,50067108,0,117464424,0,300830572,0,709098696
%N A047760 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type F.
%C A047760 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 F 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 4. An achiral polyomino is identical to its reflection. - _Robert A. Russell_, Mar 22 2024
%H A047760 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 A047760 Robert A. Russell, <a href="/A047760/a047760.txt">Mathematica Graphics3D program for A047760 examples</a>
%F A047760 If n=2m+1 then (1/2)*(A047750(m) - A047753(n) - A047751(n)), otherwise 0.
%F A047760 G.f.: (2*(G(z^4)-1)/z - z*G(z^4) + 2z*G(z^4)^2 - z*G(z^8) - z^5*G(z^8)^2) / 2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - _Robert A. Russell_, Mar 22 2024
%t A047760 Table[Switch[Mod[n,4],3,6Binomial[(3n-1)/4,(n+1)/2]/(n+3),1,(5n-1)Binomial[(3n-3)/4,(n-1)/4]/((n+1)(n+3))-Switch[Mod[n,8],1,2Binomial[(3n-3)/8,(n-1)/8]/(n+3),5,2Binomial[(3n-7)/8,(n+3)/8]/(n-1),_,0],_,0],{n,60}] (* _Robert A. Russell_, Mar 22 2024 *)
%Y A047760 Cf. A047761, A047750.
%Y A047760 Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047751 (type K), A047753 (type I).
%K A047760 nonn
%O A047760 1,7
%A A047760 _N. J. A. Sloane_