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A047773 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type D.

%I A047773 #26 Apr 04 2024 10:22:42
%S A047773 0,0,0,1,0,0,1,1,0,2,3,0,3,5,0,7,11,0,12,23,0,30,55,0,55,114,0,143,
%T A047773 272,0,273,588,0,728,1428,0,1428,3156,0,3876,7750,0,7752,17427,0,
%U A047773 21318,43263,0,43263,98516,0,120175,246672,0,246675,567281,0
%N A047773 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type D.
%C A047773 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 D achiral symmetry and n tetrahedral cells. The center of symmetry is the altitude of a tetrahedral cell (32); the order of the symmetry group is 6. An achiral polyomino is identical to its reflection. - _Robert A. Russell_, Mar 23 2024
%H A047773 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 A047773 Robert A. Russell, <a href="/A047773/a047773.txt">Mathematica Graphics3D program for A047773 examples</a>
%F A047773 If n=3m+2 then (1/2)*(A047750(m) - 2*A047751(n) - A047764(n)), if n=3m+1 then A047749(m), otherwise 0.
%F A047773 G.f.: (G(z^6)-1)/z + z*G(z^6) - z + z^2*G(z^6)^2 + z^4*G(z^6)^2 - z^5*G(z^24) - z^17*G(z^24)^2 - (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 23 2024
%t A047773 Table[Switch[Mod[n,6],1,If[1==n,0,3Binomial[(n-1)/2,(n-1)/6]/(n+2)],2,6Binomial[n/2,(n-2)/6]/(n+4)-3Binomial[(n-2)/2,(n-2)/6]/(2n+2)-If[2==Mod[n,12],3Binomial[(n-2)/4,(n-2)/12],6Binomial[(n-4)/4,(n-8)/12]]/(n+4),4,6Binomial[(n-2)/2,(n-4)/6]/(n+2),5,3Binomial[(n+1)/2,(n+1)/6]/(n+4)-Switch[Mod[n,24],5,12Binomial[(n-5)/8,(n-5)/24],17,24Binomial[(n-9)/8,(n-17)/24],_,0]/(n+7),_,0],{n,60}] (* _Robert A. Russell_, Mar 23 2024 *)
%o A047773 (PARI) /* here U=A047749, V=A047750, K=A047751, and Q=A047764 */
%o A047773 U(n)={if(n%2,my(m=(n-1)/2);(3*m+1)!/((m+1)!*(2*m+1)!),my(m=n/2);(3*m)!/(m!*(2*m+1)!))};
%o A047773 V(n)={if(n%2,my(m=(n-1)/2);6*(3*m+2)!/(m!*(2*m+3)!),my(m=n/2);(3*m)!*(5*m+1)/((m+1)!*(2*m+1)!))};
%o A047773 K(n)={if(n==1,1,if(n<5,0,if(n%12==5,U((n-5)/12),0)))};
%o A047773 Q(n)={if(n<8,0,if(n%6==2,U((n-2)/6),0))};
%o A047773 D(n)={if(n<3||n%3==0,0,if(n%3==1,U((n-1)/3),(1/2)*(V((n-2)/3)-2*K(n)-Q(n))))};
%o A047773 for(k=1,57,print1(D(k),", ")) \\ _Hugo Pfoertner_, Mar 07 2020
%Y A047773 Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U), A047750 (type V), A047751 (type K), A047764 (type Q).
%K A047773 nonn
%O A047773 1,10
%A A047773 _N. J. A. Sloane_