OFFSET
1,10
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
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
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 *)
PROG
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
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved