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

Author

Robert A. Russell, Sep 28 2020

Keywords

• nonn

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}.

Examples

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

Links

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

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).

Programs

Mathematica

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

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.