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

1,3

Comments

Each member of a chiral pair is a reflection, but not a rotation, of the other.

Links

Table of n, a(n) for n=1..24.

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

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).

Example

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.

Mathematica

Table[(n-1)n^2(n+1)(8-5n^2+n^4)/48, {n, 30}]
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {0, 1, 66, 920, 6350, 29505, 106036, 317856, 832140}, 30] (* Harvey P. Dale, Nov 21 2025 *)

Prog

(PARI) a(n)=(n^8-6*n^6+13*n^4-8*n^2)/48 \\ Charles R Greathouse IV, May 20 2026

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).

Sequence in context: A101093 A293613 A304838 * A056468 A027785 A271757

Adjacent sequences: A337893 A337894 A337895 * A337897 A337898 A337899

Keyword

nonn,easy

Author

Robert A. Russell, Sep 28 2020

Status

approved