A337959 Number of chiral pairs of colorings of the 30 triangular faces of a regular icosahedron or the 30 vertices of a regular dodecahedron using n or fewer colors.

0, 8388, 28998090, 9160633008, 794699283870, 30467722237092, 664933856235516, 9607670743188672, 101313843935748516, 833333209516666980, 5606249568529546134, 31947998829845093424, 158374695227965468434

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

• Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Crossrefs

Cf. A054472 (oriented), A252704 (unoriented), A337960 (achiral).

Other elements: A337964 (edges), A337961 (dodecahedron faces, icosahedron vertices).

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

Programs

Mathematica

Table[(n^20-15n^12+14n^10+20n^8+4n^4-24n^2)/120,{n,30}]

Formula

a(n) = (n-1) * n^2 * (n+1) * (n^2+2) * (n^14 - n^12 + 3*n^10 - 5*n^8 - 4*n^6 + 8*n^4 + 4*n^2 + 12) /120.

a(n) = 8388*C(n,2) + 28972926*C(n,3) + 9044690976*C(n,4) + 749186015850*C(n,5) + 25836356193012*C(n,6) + 468028878138864*C(n,7) + 5097432576698784*C(n,8) + 36322117709159520*C(n,9) + 178947768558202560*C(n,10) + 632296225414909440*C(n,11) + 1640646875114311680*C(n,12) + 3168965153453299200*C(n,13) + 4578694359419980800*C(n,14) + 4929160839482880000*C(n,15) + 3897035952819609600*C(n,16) + 2197214626134528000*C(n,17) + 836310065310720000*C(n,18) + 192604742313984000*C(n,19) + 20274183401472000*C(n,20), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.

a(n) = A054472(n) - A252704(n) = (A054472(n) - A337960(n)) / 2 = A252704(n) - A337960(n).