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A337960 Number of achiral colorings of the 30 triangular faces of a regular icosahedron or the 30 vertices of a regular dodecahedron using n or fewer colors. 7
1, 1048, 133875, 4211872, 61198135, 545203800, 3465030541, 17197766272, 70665499413, 250166670040, 785039389519, 2230057075104, 5826818931739, 14178299017624, 32446195329465, 70387069393408, 145689159233737 (list; graph; refs; listen; history; text; internal format)
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 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_2^10
Edge rotation* 15 x_1^4x_2^8 Asterisk indicates that the
Vertex rotation* 20 x_2^1x_6^3 operation is followed by an
Small face rotation* 12 x_10^2 inversion.
Large face rotation* 12 x_10^2
LINKS
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).
FORMULA
a(n) = n^2 * (15*n^10 + n^8 + 20*n^2 + 24) / 60.
a(n) = 1*C(n,1) + 1046*C(n,2) + 130734*C(n,3) + 3682656*C(n,4) + 41467050*C(n,5) + 238531284*C(n,6) + 791012880*C(n,7) + 1603496160*C(n,8) + 2021060160*C(n,9) + 1546836480*C(n,10) + 658627200*C(n,11) + 119750400*C(n,12), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A252704(n) - A054472(n) = A054472(n) - 2*A337959(n) = A252704(n) - A337959(n).
MATHEMATICA
Table[(15n^12+n^10+20n^4+24n^2)/60, {n, 30}]
CROSSREFS
Cf. A054472 (oriented), A252704 (unoriented), A337959 (chiral).
Other elements: A337953 (edges), A337962 (dodecahedron faces, icosahedron vertices).
Other polyhedra: A006003 (tetrahedron), A337898 (cube faces, octahedron vertices), A337897 (octahedron faces, cube vertices).
Sequence in context: A236273 A251375 A281033 * A331346 A164771 A031600
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 03 2020
STATUS
approved

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Last modified July 12 23:13 EDT 2024. Contains 374257 sequences. (Running on oeis4.)