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Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type C.
6

%I #31 Apr 28 2024 09:50:48

%S 0,0,0,0,0,0,1,1,0,5,6,0,26,32,0,133,176,0,708,952,0,3861,5302,0,

%T 21604,29960,0,123266,172535,0,715221,1007575,0,4206956,5959656,0,

%U 25032840,35622384,0,150413348,214875099,0,911379384,1306303424,0,5562367173

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

%C 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

%H L. W. Beineke and R. E. Pippert, <a href="https://doi.org/10.4153/CJM-1974-006-x">Enumerating dissectable polyhedra by their automorphism groups</a>, Canad. J. Math., 26 (1974), 50-67.

%H Robert A. Russell, <a href="/A047774/a047774.txt">Mathematica Graphics3D program for A047774 examples</a>.

%F From _Robert A. Russell_, Mar 25 2024: (Start)

%F 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.

%F 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)

%p # T=A001764

%p T := proc(n)

%p if n < 0 then

%p 0;

%p else

%p (3*n)!/n!/(2*n+1)! ;

%p end if;

%p end proc:

%p # U=A047749

%p U := proc(n)

%p if type(n,'integer') then

%p if type(n,'even') then

%p T(n/2) ;

%p else

%p (3*n-1)/(n+1)*T((n-1)/2) ;

%p end if;

%p else

%p 0 ;

%p end if;

%p end proc:

%p # V=A047750

%p V := proc(n)

%p if type(n,'integer') then

%p if type(n,'even') then

%p 2*U(n+1)-U(n) ;

%p else

%p 2*U(n+1) ;

%p end if;

%p else

%p 0;

%p end if;

%p end proc:

%p K := proc(n)

%p if n < 1 then

%p 0 ;

%p elif n = 1 then

%p 1;

%p else

%p U((n-5)/12) ;

%p end if;

%p end proc:

%p J := proc(n)

%p if type((n-5)/12,'integer') then

%p T((n-5)/12)-K(n) ;

%p %/2 ;

%p else

%p 0;

%p end if ;

%p end proc:

%p Q := proc(n)

%p if type((n-2)/6,'integer') then

%p U((n-2)/6) ;

%p else

%p 0 ;

%p end if;

%p end proc:

%p N := proc(n)

%p if type((n-2)/6,'integer') then

%p T((n-2)/6)-Q(n) ;

%p %/2 ;

%p else

%p 0;

%p end if ;

%p end proc:

%p DD := proc(n)

%p 2*U((n-1)/3)+V((n-2)/3)-2*K(n)-Q(n) ;

%p %/2 ;

%p end proc:

%p OO := proc(n)

%p if type((n-2)/6,'integer') then

%p T((n-2)/6)-Q(n) ;

%p %/2 ;

%p else

%p 0;

%p end if ;

%p end proc:

%p C := proc(n)

%p if n = 1 then

%p 0;

%p elif modp(n,3) = 1 then

%p T((n-1)/3)-DD(n) ;

%p %/2 ;

%p else

%p U((2*n-1)/3)-2*DD(n)-4*J(n) -2*K(n)-2*N(n)-2*OO(n)-Q(n) ;

%p %/4 ;

%p end if;

%p end proc:

%p seq(C(n),n=1..50) ; # _R. J. Mathar_, Jul 10 2013

%t t[n_?IntegerQ] := Binomial[3 n, n] / (2 n + 1); t[_] = 0;

%t u[n_] := t[n/2] + ((3n-1)/(n+1)) t[(n-1)/2];

%t 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;

%t Array[c, 46] (* _Andrey Zabolotskiy_, Jul 30 2023 and Apr 03 2024, using _R. J. Mathar_'s code above *)

%t 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 *)

%Y 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).

%K nonn,easy

%O 1,10

%A _N. J. A. Sloane_

%E More terms from _R. J. Mathar_, Jul 10 2013