A337960 Number of achiral colorings of the 30 triangular faces of a regular icosahedron or the 30 vertices of a regular dodecahedron using n or fewer colors.

1, 1048, 133875, 4211872, 61198135, 545203800, 3465030541, 17197766272, 70665499413, 250166670040, 785039389519, 2230057075104, 5826818931739, 14178299017624, 32446195329465, 70387069393408, 145689159233737

Offset: 1

Author

Robert A. Russell, Oct 03 2020

Keywords

• nonn
• easy

Comments

An achiral coloring is identical to its reflection. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.

There are 60 elements in the automorphism group of the regular dodecahedron/icosahedron that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the 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_2^10

Edge rotation* 15 x_1^4x_2^8 Asterisk indicates that the

Vertex rotation* 20 x_2^1x_6^3 operation is followed by an

Small face rotation* 12 x_10^2 inversion.

Large face rotation* 12 x_10^2

Links

• Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Crossrefs

Cf. A054472 (oriented), A252704 (unoriented), A337959 (chiral).

Other elements: A337953 (edges), A337962 (dodecahedron faces, icosahedron vertices).

Other polyhedra: A006003 (tetrahedron), A337898 (cube faces, octahedron vertices), A337897 (octahedron faces, cube vertices).

Programs

Mathematica

Table[(15n^12+n^10+20n^4+24n^2)/60,{n,30}]

Formula

a(n) = n^2 * (15*n^10 + n^8 + 20*n^2 + 24) / 60.

a(n) = 1*C(n,1) + 1046*C(n,2) + 130734*C(n,3) + 3682656*C(n,4) + 41467050*C(n,5) + 238531284*C(n,6) + 791012880*C(n,7) + 1603496160*C(n,8) + 2021060160*C(n,9) + 1546836480*C(n,10) + 658627200*C(n,11) + 119750400*C(n,12), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.

a(n) = 2*A252704(n) - A054472(n) = A054472(n) - 2*A337959(n) = A252704(n) - A337959(n).