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
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.
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
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 03 2020
STATUS
approved