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A371351
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Number of achiral polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}.
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14
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1, 1, 1, 2, 4, 8, 15, 37, 73, 182, 364, 952, 1944, 5169, 10659, 28842, 60115, 164450, 345345, 953814, 2016144, 5609760, 11920740, 33378072, 71250060, 200553733, 429757960, 1215177680, 2612635888, 7416503776
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OFFSET
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1,4
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COMMENTS
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Also number of achiral simplicial 3-clusters or stack polytopes with n tetrahedral cells. An achiral polyomino is identical to its reflection.
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LINKS
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FORMULA
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a(n) = ([0==n mod 2]*2*C(3n/2,n) + [1==n mod 2]*3*C((3n-1)/2,n) + [1==n mod4]*3*C((3n-3)/4,(n-1)/2) + [2==n mod6]*3*C(n/2-1,(n-2)/3)) / (3n+3).
a(n) = 2*H(3,n) - h(3,n) in Table 8 of Hering link.
G.f.: (-4 + 4*G(z^2) + 3z*G(z^2)^2 + 3z*G(z^4) + 2z^2*G(z^6)) / 6, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764.
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MATHEMATICA
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Table[(If[OddQ[n], 3Binomial[(3n-1)/2, n], 2Binomial[3n/2, n]]+If[1==Mod[n, 4], 3Binomial[(3n-3)/4, (n-1)/2], 0]+If[2==Mod[n, 6], 3Binomial[n/2-1, (n-2)/3], 0])/(3n+3), {n, 30}]
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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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