OFFSET
1,2
COMMENTS
Each chiral pair is counted as two when enumerating oriented arrangements. Also called a 5-cell or pentachoron. The Schläfli symbol is {3,3,3}, and it has 5 tetrahedral facets (vertices).
There are 60 elements in the rotation group of the 4-dimensional simplex. Each is an even permutation of the vertices and can be associated with a partition of 5 based on the conjugacy class of the permutation. The first formula is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Partition Count Even Cycle Indices
5 24 x_5^1
311 20 x_1^2x_3^1
221 15 x_1^1x_2^2
11111 1 x_1^5
LINKS
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
FORMULA
a(n) = n*(24 + 35*n^2 + n^4)/60.
a(n) = binomial(4+n,5) + binomial(n,5).
a(n) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4) + 2*C(n,5), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
From Elmo R. Oliveira, Aug 31 2025: (Start)
G.f.: x*(1 + x^4)/(1 - x)^6.
E.g.f.: x*(60 + 120*x + 60*x^2 + 10*x^3 + x^4)*exp(x)/60.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 6. (End)
EXAMPLE
For a(2)=6, the colors are AAAAA, AAAAB, AAABB, AABBB, ABBBB, and BBBBB.
MATHEMATICA
Table[n (24 + 35 n^2 + n^4)/60, {n, 40}]
PROG
(PARI) my(x='x+O('x^39)); Vec(x*(x^4+1)/(x-1)^6) \\ Elmo R. Oliveira, Aug 31 2025
CROSSREFS
Cf. A000389(n+4) (unoriented), A000389 (chiral), A132366(n-1) (achiral), A331350 (edges, faces), A337952 (8-cell vertices, 16-cell facets), A337956 (16-cell vertices, 8-cell facets), A338948 (24-cell), A338964 (120-cell, 600-cell).
Row 4 of A324999 (oriented colorings of facets or vertices of an n-simplex).
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Sep 28 2020
EXTENSIONS
More terms from Elmo R. Oliveira, Aug 31 2025
STATUS
approved