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

%I #25 Apr 28 2024 09:50:42

%S 0,0,0,0,1,0,6,0,32,0,176,0,952,0,5302,0,29960,0,172536,0,1007575,0,

%T 5959656,0,35622384,0,214875104,0,1306303424,0,7995896502,0,

%U 49236826080,0,304799714960,0,1895785216039,0,11841367945110,0,74245791718824

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

%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 E chiral symmetry and n tetrahedral cells. The axis of symmetry connects opposite edge centers of a tetrahedron (31); the order of the symmetry group is 2. Each member of a chiral pair is a reflection but not a rotation of the other. - _Robert A. Russell_, Mar 22 2024

%H L. W. Beineke and R. E. Pippert, <a href="http://dx.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="/A047762/a047762.txt">Mathematica Graphics3D program for A047762 example</a>.

%F If n=2m+1 then (1/4)*(A047749(n) - 2*A047760(n) - 6*A047758(n) - 2*A047754(n) - 3*A047753(n) - 2*A047752(n) - A047751(n)), otherwise 0.

%F G.f.: (z^2*G(z^2)^2 - (2+z^2)*G(z^4) - 2*z^2*G(z^4)^2 + 2*(1 + z^2*G(z^8) + z^6*G(z^8)^2)) / (4*z), where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - _Robert A. Russell_, Mar 22 2024

%t Table[If[OddQ[n],Binomial[(3n-1)/2,(n-1)/2]/(n+1)-If[1==Mod[n,4],Binomial[(3n-3)/4,(n-1)/4]/((n+1))+(4Binomial[(3n+1)/4,(n-1)/4]-If[1==Mod[n,8],4Binomial[(3n-3)/8,(n-1)/8],8Binomial[(3n-7)/8,(n-5)/8]])/(n+3),2Binomial[(3n+3)/4,(n+1)/4]/(n+3)],0]/2,{n,40}] (* _Robert A. Russell_, Mar 22 2024 *)

%Y Cf. A047749, A047763.

%Y Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047751 (type K), A047752 (type J), A047753 (type I), A047754 (type H), A047758 (type G), A047760 (type F).

%K nonn

%O 1,7

%A _N. J. A. Sloane_