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

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, 272, 0, 273, 588, 0, 728, 1428, 0, 1428, 3156, 0, 3876, 7750, 0, 7752, 17427, 0, 21318, 43263, 0, 43263, 98516, 0, 120175, 246672, 0, 246675, 567281, 0

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Links

• L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
• Robert A. Russell, Mathematica Graphics3D program for A047773 examples

Crossrefs

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U), A047750 (type V), A047751 (type K), A047764 (type Q).

Programs

Mathematica

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 *)

PARI

/* here U=A047749, V=A047750, K=A047751, and Q=A047764 */
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)!))};
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)!))};
K(n)={if(n==1,1,if(n<5,0,if(n%12==5,U((n-5)/12),0)))};
Q(n)={if(n<8,0,if(n%6==2,U((n-2)/6),0))};
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))))};
for(k=1,57,print1(D(k),", ")) \\ Hugo Pfoertner, Mar 07 2020

Formula

If n=3m+2 then (1/2)*(A047750(m) - 2*A047751(n) - A047764(n)), if n=3m+1 then A047749(m), otherwise 0.

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