cp's OEIS Frontend

A337896 Number of chiral pairs of colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

%I A337896 #6 Mar 09 2024 11:15:02
%S A337896 0,1,66,920,6350,29505,106036,317856,832140,1961025,4248310,8590296,
%T A337896 16398746,29814785,51983400,87399040,142333656,225359361,347978730,
%U A337896 525376600,777308070,1129138241,1613050076,2269437600
%N A337896 Number of chiral pairs of colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.
%C A337896 Each member of a chiral pair is a reflection, but not a rotation, of the other.
%H A337896 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).
%F A337896 a(n) = (n-1) * n^2 * (n+1) * (8 - 5*n^2 + n^4) / 48.
%F A337896 a(n) = 1*C(n,2) + 63*C(n,3) + 662*C(n,4) + 2400*C(n,5) + 3900*C(n,6) + 2940*C(n,7) + 840*C(n,8), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
%F A337896 G.f.: x^2 * (1+x) * (1+56*x+306*x^2+56*x^3+x^4) / (1-x)^9.
%F A337896 a(n) = A000543(n) - A128766(n) = (A000543(n) - A337897(n)) / 2 = A128766(n) - A337897(n).
%e A337896 For a(2)=1, centering the octahedron (cube) at the origin and aligning the diagonals (edges) with the axes, color the faces (vertices) in the octants ---, --+, -++, and +++ with one color and the other 4 elements with the other color.
%t A337896 Table[(n-1)n^2(n+1)(8-5n^2+n^4)/48, {n,30}]
%Y A337896 Cf. A000543 (oriented), A128766(unoriented), A337897 (achiral).
%Y A337896 Other elements: A337406 (edges), A093566(n+1) (cube faces, octahedron vertices).
%Y A337896 Other polyhedra: A000332 (simplex), A093566(n+1) (cube/octahedron).
%Y A337896 Row 3 of A325014 (chiral pairs of colorings of orthoplex facets or orthotope vertices).
%Y A337896 Row 3 of A337893 (chiral pairs of colorings of orthoplex faces or orthotope peaks).
%K A337896 nonn
%O A337896 1,3
%A A337896 _Robert A. Russell_, Sep 28 2020