A337899 Number of chiral pairs of colorings of the edges of a regular tetrahedron using n or fewer colors.

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

Offset

1,3

Comments

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}.

Links

Table of n, a(n) for n=1..28.

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Formula

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.

Example

For a(2)=1, two opposite edges and one edge connecting those have one color; the other three edges have the other color.

Mathematica

Table[(n-1)n^2(n+1)(n^2-2)/24, {n, 40}]

Crossrefs

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: A220388 A220151 A372751 * A200987 A107731 A003702

Adjacent sequences: A337896 A337897 A337898 * A337900 A337901 A337902

Keyword

nonn

Author

Robert A. Russell, Sep 28 2020

Status

approved