A337962 Number of achiral colorings of the 12 pentagonal faces of a regular dodecahedron or the 12 vertices of a regular icosahedron using n or fewer colors.

1, 68, 1659, 16464, 97935, 420708, 1443197, 4198720, 10770597, 25016740, 53619335, 107545296, 204013251, 369072900, 640912665, 1074021632, 1744341865, 2755557252, 4246675123, 6401066960, 9457144599, 13720858404

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 dodecahedron face (icosahedron 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_2^6

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

Vertex rotation* 20 x_6^2 operation is followed by an

Small face rotation* 12 x_2^1x_10^1 inversion.

Large face rotation* 12 x_2^1x_10^1

Links

Table of n, a(n) for n=1..22.

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

a(n) = n^2 * (15*n^6 + n^4 + 44)/60.

a(n) = 1*C(n,1) + 66*C(n,2) + 1458*C(n,3) + 10232*C(n,4) + 31530*C(n,5) + 47892*C(n,6) + 35280*C(n,7) + 10080*C(n,8), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.

a(n) = 2*A252705(n) - A000545(n) = A000545(n) - 2*A337961(n) = A252705(n) - A337961(n).

From Stefano Spezia, Oct 04 2020: (Start)

G.f.: x*(1+59*x+1083*x^2+3897*x^3+3087*x^4+1083*x^5+59*x^6+x^7)/(1-x)^9.

a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-8) for n > 8.

(End)

Mathematica

Table[(15n^8+n^6+44n^2)/60, {n, 30}]

Crossrefs

Cf. A000545 (oriented), A252705 (unoriented), A337961 (chiral).

Other elements: A337960 (dodecahedron vertices, icosahedron faces), A337953 (edges).

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

Sequence in context: A245875 A329444 A230687 * A231106 A228332 A220722

Adjacent sequences: A337959 A337960 A337961 * A337963 A337964 A337965

Keyword

nonn,easy

Author

Robert A. Russell, Oct 03 2020

Status

approved