cp's OEIS Frontend

A047753 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type I.

%I A047753 #23 Apr 30 2024 07:51:27
%S A047753 0,0,0,0,0,0,0,0,1,0,0,0,2,0,0,0,2,0,0,0,7,0,0,0,12,0,0,0,29,0,0,0,55,
%T A047753 0,0,0,143,0,0,0,271,0,0,0,728,0,0,0,1428,0,0,0,3873,0,0,0,7752,0,0,0,
%U A047753 21318,0,0,0,43256,0,0,0,120175,0,0,0,246675,0,0,0
%N A047753 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type I.
%C A047753 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 I achiral symmetry and n tetrahedral cells. The center of symmetry is the center of a tetrahedral cell (3.1); the order of the symmetry group is 8. An achiral polyomino is identical to its reflection. - _Robert A. Russell_, Mar 22 2024
%H A047753 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 A047753 Robert A. Russell, <a href="/A047753/a047753.txt">Mathematica Graphics3D program for A047753 example</a>.
%F A047753 If n=4m+1 then A047749(m) - A047751(n), otherwise 0.
%F A047753 G.f.: z*G(z^8) + z^5*G(z^8)^2 - z - z^5*G(z^24) - z^17*G(z^24)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - _Robert A. Russell_, Mar 22 2024
%t A047753 Table[Switch[Mod[n,8],1,4Binomial[(3n-3)/8,(n-1)/8]/(n+3)-If[17==Mod[n,24],24Binomial[(n-9)/8,(n-17)/24]/(n+7),0],5,4Binomial[(3n-7)/8,(n+3)/8]/(n-1)-If[5==Mod[n,24],12Binomial[(n-5)/8,(n-5)/12]/(n+7),0],_,0]-Boole[1==n],{n,50}] (* _Robert A. Russell_, Mar 22 2024 *)
%Y A047753 Cf. A047767.
%Y A047753 Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047751 (type K), A047749 (type U).
%K A047753 nonn
%O A047753 1,13
%A A047753 _N. J. A. Sloane_