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.

0, 1, 66, 920, 6350, 29505, 106036, 317856, 832140, 1961025, 4248310, 8590296, 16398746, 29814785, 51983400, 87399040, 142333656, 225359361, 347978730, 525376600, 777308070, 1129138241, 1613050076, 2269437600

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.

Examples

			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.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Crossrefs

Cf. A000543 (oriented), A128766(unoriented), A337897 (achiral).

Other elements: A337406 (edges), A093566(n+1) (cube faces, octahedron vertices).

Other polyhedra: A000332 (simplex), A093566(n+1) (cube/octahedron).

Row 3 of A325014 (chiral pairs of colorings of orthoplex facets or orthotope vertices).

Row 3 of A337893 (chiral pairs of colorings of orthoplex faces or orthotope peaks).

Programs

Mathematica

Table[(n-1)n^2(n+1)(8-5n^2+n^4)/48, {n,30}]

Formula

a(n) = (n-1) * n^2 * (n+1) * (8 - 5*n^2 + n^4) / 48.

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.

G.f.: x^2 * (1+x) * (1+56*x+306*x^2+56*x^3+x^4) / (1-x)^9.

a(n) = A000543(n) - A128766(n) = (A000543(n) - A337897(n)) / 2 = A128766(n) - A337897(n).