A338966 Number of chiral pairs of colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using subsets of a set of n colors.

92307499707128546879177569498768, 124792381938502167387269721273817892704188259502965515, 122697712831832245109951209382504597654581237223625701047064169830144

Offset: 2

Author

Robert A. Russell, Dec 04 2020

Keywords

• nonn
• easy

Comments

Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual.

Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide formulas here.

For the 600 facets of the 600-cell (vertices of the 120-cell), the formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 - 816*n^60 - 1440*n^66 + 40*n^100 - 800*n^104 - 1200*n^114 + 624*n^120 + 60*n^150 - 1800*n^152 + 40*n^200 + 400*n^208 - 59*n^300 + 450*n^302 - 60*n^330 + n^600) / 14400.

For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the formula is (960 n^24 + 1440 n^36 + 960 n^48 + 1200 n^60 + 336 n^72 - 1152 n^76 - 1440 n^84 - 760 n^120 - 1200 n^132 + 336 n^144 + 288 n^152 + 60 n^180 - 1800 n^182 + 440 n^240 - 59 n^360 + 450 n^364 - 60 n^396 + n^720) / 14400.

For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 - 816*n^120 - 1440*n^128 + 40*n^200 - 800*n^202 - 1200*n^216 + 624*n^240 + 60*n^300 - 1800*n^302 + 40*n^400 + 400*n^404 - 59*n^600 + 450*n^604 - 60*n^640 + n^1200) / 14400.

Links

• Robert A. Russell, Table of n, a(n) for n = 2..30
• Index entries for linear recurrences with constant coefficients, order 121.

Crossrefs

Cf. A338964 (oriented), A338965 (unoriented), A338967 (achiral), A338982 (exactly n colors), A000389 (5-cell), A337954 (8-cell vertices, 16-cell facets), A234249 (16-cell vertices, 8-cell facets), A338950 (24-cell).

Programs

Mathematica

Table[(960n^4 +1440n^6 +960n^8 +1200n^10 +336n^12 +288n^16 -1440n^17 -1440n^19 +40n^20 +400n^22 -1200n^23 +336n^24 -1200n^27 +60n^30 -1800n^31 +288n^32 +40n^40 +400n^44 +n^60 -60n^61 +450n^62 -60n^75 +n^120)/14400, {n,2,10}]

Formula

a(n) = (960*n^4 + 1440*n^6 + 960*n^8 + 1200*n^10 + 336*n^12 + 288*n^16 - 1440*n^17 - 1440*n^19 + 40*n^20 + 400*n^22 - 1200*n^23 + 336*n^24 - 1200*n^27 + 60*n^30 - 1800*n^31 + 288*n^32 + 40*n^40 + 400*n^44 + n^60 - 60*n^61 + 450*n^62 - 60*n^75 + n^120) / 14400.

a(n) = Sum_{j=2..Min(n,120)} A338982(n) * binomial(n,j).

a(n) = A338964(n) - A338965(n) =(A338964(n) - A338967(n)) / 2 = A338965(n) - A338967(n).