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

1,2

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 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

Table of n, a(n) for n=1..15.

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).

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).

Mathematica

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

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).

Sequence in context: A170798 A043628 A205410 * A210720 A233751 A205212

Adjacent sequences: A337950 A337951 A337952 * A337954 A337955 A337956

Keyword

nonn,easy

Author

Robert A. Russell, Oct 03 2020

Status

approved