A338965 Number of unoriented 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.

1, 92307499707443390526727850063504, 124792381938502167392338612231208163827413085862945471, 122697712831832245109951221276235414511846772206539032522116543043328

Offset: 1

Author

Robert A. Russell, Dec 04 2020

Keywords

• nonn
• easy

Comments

Each chiral pair is counted as one when enumerating unoriented arrangements. 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 + 2064*n^60 + 1440*n^66 + 40*n^100 + 1600*n^104 + 1200*n^114 + 624*n^120 + 60*n^150 + 1800*n^152 + 40*n^200 + 400*n^208 + 61*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 + 1728 n^76 + 1440 n^84 + 1640 n^120 + 1200 n^132 + 336 n^144 + 288 n^152 + 60 n^180 + 1800 n^182 + 440 n^240 + 61 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 + 2064*n^120 + 1440*n^128 + 40*n^200 + 1600*n^202 + 1200*n^216 + 624*n^240 + 60*n^300 + 1800*n^302 + 40*n^400 + 400*n^404 + 61*n^600 + 450*n^604 + 60*n^640 + n^1200) / 14400.

Links

• Robert A. Russell, Table of n, a(n) for n = 1..30
• Index entries for linear recurrences with constant coefficients, order 121.

Crossrefs

Cf. A338964 (oriented), A338966 (chiral), A338967 (achiral), A338981 (exactly n colors), A000389 (5-cell), A128767 (8-cell vertices, 16-cell facets), A337957(16-cell vertices, 8-cell facets), A338949 (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,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=1..Min(n,120)} A338981(n) * binomial(n,j).

a(n) = A338964(n) - A338966(n) =(A338964(n) + A338967(n)) / 2 = A338966(n) + A338967(n).