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A337899
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Number of chiral pairs of colorings of the edges of a regular tetrahedron using n or fewer colors.
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2
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0, 1, 21, 140, 575, 1785, 4606, 10416, 21330, 40425, 71995, 121836, 197561, 308945, 468300, 690880, 995316, 1404081, 1943985, 2646700, 3549315, 4694921, 6133226, 7921200, 10123750, 12814425, 16076151, 20001996
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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. A regular tetrahedron has 6 edges and Schläfli symbol {3,3}.
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LINKS
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Table of n, a(n) for n=1..28.
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FORMULA
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a(n) = (n-1) * n^2 * (n+1) * (n^2-2) / 24.
a(n) = 1*C(n,2) + 18*C(n,3) + 62*C(n,4) + 75*C(n,5) + 30*C(n,6), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A046023(n) - A063842(n-1) = (A046023(n) - A037270(n)) / 2 = A063842(n-1) - A037270(n).
G.f.: x^2 * (1+x) * (1+13x+x^2)/(1-x)^7.
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EXAMPLE
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For a(2)=1, two opposite edges and one edge connecting those have one color; the other three edges have the other color.
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MATHEMATICA
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Table[(n-1)n^2(n+1)(n^2-2)/24, {n, 40}]
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CROSSREFS
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Cf. A046023(unoriented), A063842(n-1) (oriented), A037270 (chiral).
Other elements: A000332 (vertices and faces).
Other polyhedra: A337406 (cube/octahedron).
Row 3 of A327085 (chiral pairs of colorings of edges or ridges of an n-simplex).
Sequence in context: A033595 A220388 A220151 * A200987 A107731 A003702
Adjacent sequences: A337896 A337897 A337898 * A337900 A337901 A337902
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KEYWORD
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nonn
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AUTHOR
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Robert A. Russell, Sep 28 2020
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STATUS
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approved
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