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%I #27 Apr 19 2024 04:35:45
%S 0,0,0,0,2,11,71,370,2005,10682,58167,320116,1789210,10121965,
%T 57933469,334919626,1953800059,11489466014,68053583772,405713887061,
%U 2433000197471,14668527134167,88869448492895,540834097467624,3304961431043989,20273201718862728,124798671079300720,770762029389852807
%N Number of chiral pairs of asymmetric dissectable polyhedra with n tetrahedral cells (type A).
%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 asymmetric (type A) with n tetrahedral cells. The order of the symmetry group is 1. Each member of a chiral pair is a reflection but not a rotation of the other. - _Robert A. Russell_, Mar 31 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="/A047776/a047776.txt">Mathematica Graphics3D program for A047776 examples</a>
%F From _Robert A. Russell_, Mar 31 2024: (Start)
%F a(n) = A001764(n)/(12(n+1)) - A047775(n)/2 - A047774(n)/3 - A047773(n)/6 - A047762(n)/2 - A047760(n)/4 - A047758(n)/4 - A047754(n)/4 - A047753(n)/8 - A047752(n)/12 - A047751(n)/24 - A047771(n)/2 - A047769(n)/2 - A047766(n)/6 - A047766(n)/6 - A047765(n)/4 - A047764(n)/12.
%F G.f.: (G(z^4) + G(z^6) - 2)/(2z) - z/3 + G(z)/6 - G(z)^2/12 + z*G(z)^4/24 - 7*G(z^2)/12 - 3z*G(z^2)^2/8 - z*G(z^3)/6 - z^2*G(z^3)^2/12 + G(z^4)/2 - z*G(z^4)/6 + (z*G(z^4)^2 + z^2*G(z^4)^2 + z*G(z^6))/2 + z^2*G(z^6)/12 + (z^2*G(z^6)^2 + z^4*G(z^6)^2 - z^2*G(z^12))/2 + z^5*G(z^12)/6 - (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)
%t Table[If[n < 5, 0, Binomial[3 n, 2 n + 2]/(3 n (n - 1))
%t - If[OddQ[n], Binomial[3 n/2 - 1/2, n + 1] 3/(n - 1),
%t 7 Binomial[3 n/2, n + 1]/(3 n)]
%t - Switch[Mod[n, 3], 1, Binomial[n - 1, 2 n/3 + 1/3]/(n - 1), 2,
%t Binomial[n - 1, 2 n/3 + 2/3]/(n - 2), _, 0]
%t + Switch[Mod[n, 4], 1, Binomial[3 n/4 - 3/4, n/2 + 1/2] 2/(3 (n - 1))
%t + Binomial[3 n/4 + 1/4, n/2 + 3/2] 4/(n - 1) +
%t Binomial[3 n/4 - 3/4, n/2 + 1/2] 4/(n + 3), 2,
%t Binomial[3 n/4 - 1/2, n/2 + 1] 8/(n - 2), 3,
%t Binomial[3 n/4 - 1/4, n/2 + 3/2] 12/(n - 3), 0,
%t Binomial[3 n/4 - 1, n/2 + 1] 12/(n - 4)] +
%t Switch[Mod[n, 6], 1, Binomial[n/2 - 1/2, n/3 + 2/3] 6/(n - 1), 2,
%t Binomial[n/2 - 1, n/3 + 1/3] 4/(n - 2) +
%t Binomial[n/2, n/3 + 4/3] 6/(n - 2) +
%t Binomial[n/2 - 1, n/3 + 1/3] 6/(n + 4), 4,
%t Binomial[n/2 - 1, n/3 + 2/3] 12/(n - 4), 5,
%t Binomial[n/2 - 1/2, n/3 + 1/3] 9/(n + 4), _, 0] +
%t Switch[Mod[n, 12], 2, -Binomial[n/4 - 1/2, n/6 + 2/3] 12/(n - 2), 5,
%t Binomial[n/4 - 5/4, n/6 - 5/6] 2/(n + 1),
%t 8, -Binomial[n/4 - 1, n/6 - 1/3] 12/(n + 4), _, 0] -
%t Switch[Mod[n, 24], 5, Binomial[n/8 - 5/8, n/12 - 5/12] 12/(n + 7), 17,
%t Binomial[n/8 - 9/8, n/12 - 5/12] 24/(n + 7), _, 0]]/2, {n, 1, 60}] (* _Robert A. Russell_, Apr 09 2012 *)
%Y Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047775 (type B), A047774 (type C). A047773 (type D), A047762 (type E), A047760 (type F), A047758 (type G), A047754 (type H), A047753 (type I), A047752 (type J), A047751 (type K), A047771 (type L), A047769 (type M), A047766 (type N|O), A047765 (type P), A047764 (type Q).
%K nonn,easy,changed
%O 1,5
%A _N. J. A. Sloane_
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