A337898 Number of achiral colorings of the 6 square faces of a cube or the 6 vertices of a regular octahedron using n or fewer colors.

1, 10, 55, 200, 560, 1316, 2730, 5160, 9075, 15070, 23881, 36400, 53690, 77000, 107780, 147696, 198645, 262770, 342475, 440440, 559636, 703340, 875150, 1079000, 1319175, 1600326, 1927485, 2306080, 2741950, 3241360

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 face (octahedron vertex) 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_2^3

Vertex rotation* 8 x_6^1 Asterisk indicates that the

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

Small face rotation* 6 x_2^1x_4^1 inversion.

Large face rotation* 3 x_1^4x_2^1

Links

Table of n, a(n) for n=1..30.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = n * (n+1) * (n+2) * (3*n^2 - 3*n + 4) / 24.

a(n) = 1*C(n,1) + 8*C(n,2) + 28*C(n,3) + 36*C(n,4) + 15*C(n,5), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.

a(n) = 2*A198833(n) - A047780(n) = A047780(n) - 2*A093566(n+1) = A198833(n) - A093566(n+1).

G.f.: x * (x + 4*x^2 + 10*x^3) / (1-x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Wesley Ivan Hurt, Sep 30 2020

Mathematica

Table[n(1+n)(2+n)(4-3n+3n^2)/24, {n, 35}]
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 10, 55, 200, 560, 1316}, 40] (* Harvey P. Dale, Feb 15 2022 *)

Prog

(PARI) a(n)=n*(n+1)*(n+2)*(3*n^2-3*n+4)/24 \\ Charles R Greathouse IV, Oct 21 2022

Crossrefs

Cf. A047780 (oriented), A198833 (unoriented), A093566(n+1) (chiral).

Other elements: A331351 (edges), A337897 (cube vertices/octahedron faces).

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

Row 3 of A325007 (orthotope facets, orthoplex vertices) and A337890 (orthotope faces, orthoplex peaks).

Sequence in context: A069155 A107352 A127761 * A357730 A341989 A341070

Adjacent sequences: A337895 A337896 A337897 * A337899 A337900 A337901

Keyword

nonn,easy

Author

Robert A. Russell, Sep 28 2020

Status

approved