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A337896
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Number of chiral pairs of colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.
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6
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0, 1, 66, 920, 6350, 29505, 106036, 317856, 832140, 1961025, 4248310, 8590296, 16398746, 29814785, 51983400, 87399040, 142333656, 225359361, 347978730, 525376600, 777308070, 1129138241, 1613050076, 2269437600
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OFFSET
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1,3
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COMMENTS
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Each member of a chiral pair is a reflection, but not a rotation, of the other.
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LINKS
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FORMULA
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a(n) = (n-1) * n^2 * (n+1) * (8 - 5*n^2 + n^4) / 48.
a(n) = 1*C(n,2) + 63*C(n,3) + 662*C(n,4) + 2400*C(n,5) + 3900*C(n,6) + 2940*C(n,7) + 840*C(n,8), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
G.f.: x^2 * (1+x) * (1+56*x+306*x^2+56*x^3+x^4) / (1-x)^9.
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EXAMPLE
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For a(2)=1, centering the octahedron (cube) at the origin and aligning the diagonals (edges) with the axes, color the faces (vertices) in the octants ---, --+, -++, and +++ with one color and the other 4 elements with the other color.
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MATHEMATICA
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Table[(n-1)n^2(n+1)(8-5n^2+n^4)/48, {n, 30}]
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CROSSREFS
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Other elements: A337406 (edges), A093566(n+1) (cube faces, octahedron vertices).
Row 3 of A325014 (chiral pairs of colorings of orthoplex facets or orthotope vertices).
Row 3 of A337893 (chiral pairs of colorings of orthoplex faces or orthotope peaks).
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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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