|
%I #9 Mar 10 2024 13:30:23
%S 0,14,3720,132184,1987720,17935806,114638048,570597216,2348263008,
%T 8320953630,26126986952,74247445272,194049316552,472265688622,
%U 1080900468480,2345089916288,4854316187136,9638888023278,18442173583176
%N Number of chiral pairs of colorings of the 12 pentagonal faces of a regular dodecahedron or the 12 vertices of a regular icosahedron using n or fewer colors.
%C Each member of a chiral pair is a reflection, but not a rotation, of the other. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).
%F a(n) = (n-1) * n^2 * (n+1) * (n^8 + n^6 - 14*n^4 + 44) / 120.
%F a(n) = 14*C(n,2) + 3678*C(n,3) + 117388*C(n,4) + 1363860*C(n,5) + 7918056*C(n,6) + 26332992*C(n,7) + 53428032*C(n,8) + 67359600*C(n,9) + 51559200*C(n,10) + 21954240*C(n,11) + 3991680*C(n,12), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
%F a(n) = A000545(n) - A252705(n) = (A000545(n) - A337962(n)) / 2 = A252705(n) - A337962(n).
%t Table[(n^12-15n^8+14n^6+44n^4-44n^2)/120,{n,30}]
%Y Cf. A000545 (oriented), A252705 (unoriented), A337962 (achiral).
%Y Other elements: A337959 (dodecahedron vertices, icosahedron faces), A337964 (edges).
%Y Other polyhedra: A000332 (tetrahedron), A093566(n+1) (cube faces, octahedron vertices), A337896 (octahedron faces, cube vertices).
%K nonn,easy
%O 1,2
%A _Robert A. Russell_, Oct 03 2020
|
