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A047762
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Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type E.
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5
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0, 0, 0, 0, 1, 0, 6, 0, 32, 0, 176, 0, 952, 0, 5302, 0, 29960, 0, 172536, 0, 1007575, 0, 5959656, 0, 35622384, 0, 214875104, 0, 1306303424, 0, 7995896502, 0, 49236826080, 0, 304799714960, 0, 1895785216039, 0, 11841367945110, 0, 74245791718824
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OFFSET
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1,7
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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