Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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