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

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 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

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

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).

Programs

Mathematica

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

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)