A047774 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type C.

0, 0, 0, 0, 0, 0, 1, 1, 0, 5, 6, 0, 26, 32, 0, 133, 176, 0, 708, 952, 0, 3861, 5302, 0, 21604, 29960, 0, 123266, 172535, 0, 715221, 1007575, 0, 4206956, 5959656, 0, 25032840, 35622384, 0, 150413348, 214875099, 0, 911379384, 1306303424, 0, 5562367173

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

One of 17 different symmetry types comprising A007173 and A027610 and one of 7 for A371350. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type C chiral symmetry and n tetrahedral cells. The axis of rotational symmetry is the altitude of a tetrahedral cell (32); the order of the symmetry group is 3. Each member of a chiral pair is a reflection but not a rotation of the other. - Robert A. Russell, Mar 25 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 A047774 examples.

Crossrefs

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047749 (type U), A047751 (type K), A047752 (type J), A047764 (type Q), A047766 (types N|O), A047773 (type D).

Programs

Maple

# T=A001764
T := proc(n)
    if n < 0 then
        0;
    else
        (3*n)!/n!/(2*n+1)! ;
    end if;
end proc:
# U=A047749
U := proc(n)
    if type(n,'integer') then
        if type(n,'even') then
            T(n/2) ;
        else
            (3*n-1)/(n+1)*T((n-1)/2) ;
        end if;
    else
        0 ;
    end if;
end proc:
# V=A047750
V := proc(n)
    if type(n,'integer') then
        if type(n,'even') then
            2*U(n+1)-U(n) ;
        else
            2*U(n+1) ;
        end if;
    else
        0;
    end if;
end proc:
K := proc(n)
    if n < 1 then
        0 ;
    elif n = 1 then
        1;
    else
        U((n-5)/12) ;
    end if;
end proc:
J := proc(n)
    if type((n-5)/12,'integer') then
        T((n-5)/12)-K(n) ;
        %/2 ;
    else
        0;
    end if ;
end proc:
Q := proc(n)
    if type((n-2)/6,'integer') then
        U((n-2)/6) ;
    else
        0 ;
    end if;
end proc:
N := proc(n)
    if type((n-2)/6,'integer') then
        T((n-2)/6)-Q(n) ;
        %/2 ;
    else
        0;
    end if ;
end proc:
DD := proc(n)
    2*U((n-1)/3)+V((n-2)/3)-2*K(n)-Q(n) ;
    %/2 ;
end proc:
OO := proc(n)
    if type((n-2)/6,'integer') then
        T((n-2)/6)-Q(n) ;
        %/2 ;
    else
        0;
    end if ;
end proc:
C := proc(n)
    if n = 1 then
        0;
      elif modp(n,3) = 1 then
        T((n-1)/3)-DD(n) ;
        %/2 ;
    else
        U((2*n-1)/3)-2*DD(n)-4*J(n) -2*K(n)-2*N(n)-2*OO(n)-Q(n) ;
        %/4 ;
    end if;
end proc:
seq(C(n),n=1..50) ; # R. J. Mathar, Jul 10 2013

Mathematica

t[n_?IntegerQ] := Binomial[3 n, n] / (2 n + 1); t[_] = 0;
u[n_] := t[n/2] + ((3n-1)/(n+1)) t[(n-1)/2];
c[n_] := (2 (t[(n-1)/3] - u[(n-1)/3] - u[(n+1)/3] + u[(n-2)/6] + u[(n-5)/12] - t[(n-5)/12]) + u[(2n-1)/3] - t[(n-2)/6]) / 4;
Array[c, 46] (* Andrey Zabolotskiy, Jul 30 2023 and Apr 03 2024, using R. J. Mathar's code above *)
Table[(If[2==Mod[n,3],3Binomial[n-1,(n-2)/3]-If[2==Mod[n,6],3Binomial[(n-2)/2,(n-2)/6],0],0]/(2n+2)-Switch[Mod[n,3],1,If[1==Mod[n,6],3Binomial[(n-1)/2,(n-1)/6],6Binomial[(n-2)/2,(n-4)/6]]/(n+2)-3Binomial[n-1,(n-1)/3]/(2n+1),2,If[2==Mod[n,6],6Binomial[n/2,(n-2)/6]-If[2==Mod[n,12],6Binomial[(n-2)/4,(n-2)/12],12Binomial[n/4-1,(n-8)/12]],3Binomial[(n+1)/2,(n+1)/6]]/(n+4),,0]-If[5==Mod[n,12],6Binomial[(n-5)/4,(n-5)/12]/(n+1)-If[5==Mod[n,24],12Binomial[(n-5)/8,(n-5)/24],24Binomial[(n-9)/8,(n-17)/24]]/(n+7),0])/2,{n,50}] (* _Robert A. Russell, Mar 25 2024 *)

Formula

From Robert A. Russell, Mar 25 2024: (Start)

a(n) = (2*A001764((n-1)/3) + A047749((2n-1)/3) - 2*A047773(n) - 4*A047752(n) - 2*A047751(n) - 4*A047766(n) - A047773(n)) / 4.

G.f.: ((1 - G(z^6))/z + z^2*(G(z^3)^2 - G(z^6))/2 + z*G(z^3) - z*G(z^6) - z^2*G(z^6)^2 - z^4*G(z^6)^2 + z^2*G(z^12) - z^5*G(z^12) + z^8*G(z^12)^2 + z^5*G(z^24) + z^17*G(z^24)^2) / 2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)

Extensions

More terms from R. J. Mathar, Jul 10 2013