A337961 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.

0, 14, 3720, 132184, 1987720, 17935806, 114638048, 570597216, 2348263008, 8320953630, 26126986952, 74247445272, 194049316552, 472265688622, 1080900468480, 2345089916288, 4854316187136, 9638888023278, 18442173583176

Offset: 1

Author

Robert A. Russell, Oct 03 2020

Keywords

• nonn
• easy

Comments

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.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Crossrefs

Cf. A000545 (oriented), A252705 (unoriented), A337962 (achiral).

Other elements: A337959 (dodecahedron vertices, icosahedron faces), A337964 (edges).

Other polyhedra: A000332 (tetrahedron), A093566(n+1) (cube faces, octahedron vertices), A337896 (octahedron faces, cube vertices).

Programs

Mathematica

Table[(n^12-15n^8+14n^6+44n^4-44n^2)/120,{n,30}]
LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{0,14,3720,132184,1987720,17935806,114638048,570597216,2348263008,8320953630,26126986952,74247445272,194049316552},20] (* Harvey P. Dale, Nov 17 2024 *)

Formula

a(n) = (n-1) * n^2 * (n+1) * (n^8 + n^6 - 14*n^4 + 44) / 120.

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.

a(n) = A000545(n) - A252705(n) = (A000545(n) - A337962(n)) / 2 = A252705(n) - A337962(n).