A337899 - OEIS

%I #6 Mar 09 2024 11:31:39

%S 0,1,21,140,575,1785,4606,10416,21330,40425,71995,121836,197561,

%T 308945,468300,690880,995316,1404081,1943985,2646700,3549315,4694921,

%U 6133226,7921200,10123750,12814425,16076151,20001996

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

%C 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}.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7, -21, 35, -35, 21, -7, 1).

%F a(n) = (n-1) * n^2 * (n+1) * (n^2-2) / 24.

%F 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.

%F a(n) = A046023(n) - A063842(n-1) = (A046023(n) - A037270(n)) / 2 = A063842(n-1) - A037270(n).

%F G.f.: x^2 * (1+x) * (1+13x+x^2)/(1-x)^7.

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

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

%Y Cf. A046023(unoriented), A063842(n-1) (oriented), A037270 (chiral).

%Y Other elements: A000332 (vertices and faces).

%Y Other polyhedra: A337406 (cube/octahedron).

%Y Row 3 of A327085 (chiral pairs of colorings of edges or ridges of an n-simplex).

%K nonn

%O 1,3

%A _Robert A. Russell_, Sep 28 2020