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

1,2

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

Table of n, a(n) for n=1..17.

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

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).

Mathematica

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

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).

Sequence in context: A236273 A251375 A281033 * A331346 A164771 A031600

Adjacent sequences: A337957 A337958 A337959 * A337961 A337962 A337963

Keyword

nonn,easy

Author

Robert A. Russell, Oct 03 2020

Status

approved