A337953 Number of achiral colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.

1, 33328, 32524281, 4312863360, 191243490675, 4239501280272, 58236754527707, 563536359633920, 4172726943804861, 25016666666700400, 126431377927701253, 554909560378102656, 2163457078062360639, 7625429483925609552, 24638829565429941975

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 edge 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_2^15

Edge rotation* 15 x_1^4x_2^13 Asterisk indicates that the

Vertex rotation* 20 x_6^5 operation is followed by an

Small face rotation* 12 x_10^3 inversion.

Large face rotation* 12 x_10^3

Links

• Index entries for linear recurrences with constant coefficients, signature (18, -153, 816, -3060, 8568, -18564, 31824, -43758, 48620, -43758, 31824, -18564, 8568, -3060, 816, -153, 18, -1).

Crossrefs

Cf. A282670 (oriented), A337963 (unoriented), A337964 (chiral).

Other elements: A337960 (dodecahedron vertices, icosahedron faces), A337962 (dodecahedron faces, icosahedron vertices).

Cf. A037270 (tetrahedron), A331351 (cube/octahedron).

Programs

Mathematica

Table[(15n^17+n^15+20n^5+24n^3)/60,{n,30}]

Formula

a(n) = n^3 * (15*n^14 + n^12 + 20*n^2 + 24) / 60.

a(n) = 1*C(n,1) + 33326*C(n,2) + 32424300*C(n,3) + 4182966200*C(n,4) + 170004083410*C(n,5) + 3156083300916*C(n,6) + 32426546302332*C(n,7) + 205938803790720*C(n,8) + 864860752435680*C(n,9) + 2503126577952000*C(n,10) + 5110943178781440*C(n,11) + 7428048096268800*C(n,12) + 7644417350169600*C(n,13) + 5446616304729600*C(n,14) + 2556525184012800*C(n,15) + 711374856192000*C(n,16) + 88921857024000*C(n,17), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.

a(n) = 2*A337963(n) - A282670(n) = A282670(n) - 2*A337964(n) = A337963(n) - A337964(n).