A337897 Number of achiral colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

1, 21, 201, 1076, 4025, 11901, 29841, 66256, 134001, 251725, 445401, 750036, 1211561, 1888901, 2856225, 4205376, 6048481, 8520741, 11783401, 16026900, 21474201, 28384301, 37055921, 47831376, 61100625, 77305501, 96944121

Offset

1,2

Comments

An achiral coloring is identical to its reflection. The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.

There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cube vertex (octahedron face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.

Conjugacy Class Count Odd Cycle Indices

Inversion 1 x_2^4

Vertex rotation* 8 x_2^1x_6^1 Asterisk indicates that the

Edge rotation* 6 x_1^4x_2^2 operation is followed by an

Small face rotation* 3 x_4^2 inversion.

Large face rotation* 6 x_2^4

Links

Table of n, a(n) for n=1..27.

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Formula

a(n) = n^2 * (7 + 2*n^2 + 3*n^4) / 12.

a(n) = 1*C(n,1) + 19*C(n,2) + 141*C(n,3) + 394*C(n,4) + 450*C(n,5) + 180*C(n,6), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.

a(n) = 2*A128766(n) - A000543(n) = A000543(n) - 2*A337896(n) = A128766(n) - A337896(n).

G.f.: x * (1+x) * (1 + 13*x + 62*x^2 + 13*x^3 + x^4) / (1-x)^7.

Mathematica

Table[n^2(7+2n^2+3n^4)/12, {n, 30}]

Crossrefs

Cf. A000543 (oriented), A128766 (unoriented), A337896 (chiral).

Other elements: A331351 (edges), A337898 (cube faces, octahedron vertices).

Other polyhedra: A006003 (tetrahedron), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).

Row 3 of A337894 (orthoplex faces, orthotope peaks) and A325015 (orthotope vertices, orthoplex facets).

Sequence in context: A159857 A221126 A239418 * A341399 A125384 A126542

Adjacent sequences: A337894 A337895 A337896 * A337898 A337899 A337900

Keyword

nonn

Author

Robert A. Russell, Sep 28 2020

Status

approved