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Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type G.
7

%I #20 Apr 30 2024 07:52:34

%S 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,8,0,0,0,42,0,0,0,232,0,0,0,1277,0,0,

%T 0,7183,0,0,0,41041,0,0,0,238315,0,0,0,1402076,0,0,0,8343804,0,0,0,

%U 50136483,0,0,0,303790544,0,0,0,1854115285,0,0,0,11388104153

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

%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 G chiral symmetry and n tetrahedral cells. The axis of symmetry is a line connecting the centers of opposite edges of a tetrahedral cell (31); the order of the symmetry group is 4. 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="/A047758/a047758.txt">Mathematica Graphics3D program for A047758 example</a>

%F If n=4m+1 then (1/6)*(A001764(m) - 3*A047753(n) - 2*A047752(n) - A047751(n)), otherwise 0.

%F G.f.: z * (2 + G(z^4) - z^4*G(z^12) - 3 * (G(z^8) + z^4*G(z^8)^2 - z^4*G(z^24) - z^16*G(z^24)^2)) / 6, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - _Robert A. Russell_, Mar 22 2024

%t Table[If[1==Mod[n,4],(2Boole[1==n]+2Binomial[(3n-3)/4,(n-1)/4]/(n+1)-If[1==Mod[n,8],12Binomial[(3n-3)/8,(n-1)/8]/(n+3),12Binomial[(3n-7)/8,(n+3)/8]/(n-1)]-If[5==Mod[n,12],6Binomial[(n-5)/4,(n-5)/12]/(n+1)-If[5==Mod[n,24],36Binomial[(n-5)/8,(n-5)/12],72Binomial[(n-9)/8,(n-17)/24]]/(n+7),0])/6,0],{n,50}] (* _Robert A. Russell_, Mar 22 2024 *)

%Y Cf. A047772.

%Y Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047751 (type K), A047752 (type J), A047753 (type I).

%K nonn

%O 1,17

%A _N. J. A. Sloane_