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

1,10

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

Table of n, a(n) for n=1..46.

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.

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)

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 *)

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).

Sequence in context: A362529 A200010 A105580 * A243108 A355186 A287610

Adjacent sequences: A047771 A047772 A047773 * A047775 A047776 A047777

Keyword

nonn,easy,changed

Author

N. J. A. Sloane

Extensions

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

Status

approved