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A047766
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Number of dissectable polyhedra with n tetrahedral cells with symmetry of type N or chiral pairs with symmetry of type O.
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9
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 26, 0, 0, 0, 0, 0, 133, 0, 0, 0, 0, 0, 708, 0, 0, 0, 0, 0, 3861, 0, 0, 0, 0, 0, 21604, 0, 0, 0, 0, 0, 123266, 0, 0, 0, 0, 0, 715221, 0, 0, 0, 0, 0, 4206956, 0, 0, 0, 0, 0, 25032840, 0, 0, 0, 0
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OFFSET
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1,20
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COMMENTS
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Two of 17 different symmetry types comprising A007173 and A027610. Type N is one of 10 for A371351; type O 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 N achiral symmetry or type O chiral symmetry and n tetrahedral cells. The center of symmetry is the center of a tetrahedral face (2.10); the order of the symmetry group is 6. For type N, the two identical rooted polyominoes sharing the central face are a chiral pair reflected in that face; for type O they have the same orientation. - 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^6) - z^2*G(z^12) - z^8*G(z^12)^2) / 2, 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[Switch[Mod[n, 12], 2, 3Binomial[(n-2)/2, (n-2)/6]/(2n+2)-3Binomial[(n-2)/4, (n-2)/12]/(n+4), 8, 3Binomial[(n-2)/2, (n-2)/6]/(2n+2)-6Binomial[(n-4)/4, (n-2)/6]/(n+4), _, 0], {n, 50}] (* 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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EXTENSIONS
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2nd A-number in the formula corrected by R. J. Mathar, Oct 21 2008
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STATUS
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approved
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