A389090 a(n) = n^4 - 4*(n^3-n)/3.

0, 1, 8, 49, 176, 465, 1016, 1953, 3424, 5601, 8680, 12881, 18448, 25649, 34776, 46145, 60096, 76993, 97224, 121201, 149360, 182161, 220088, 263649, 313376, 369825, 433576, 505233, 585424, 674801, 774040, 883841, 1004928, 1138049, 1283976, 1443505, 1617456, 1806673, 2012024, 2234401, 2474720

Offset

0,3

Comments

Number of unit complete 16-cells contained in an n-scale 16-cell composed of a 16-cell honeycomb. The number of halved 16-cells in it is 8*(n^3-n)/3 = A102860(n+1).

Assume that the n-scale 16-cell has vertices (+-n,0,0,0), (0,+-n,0,0), (0,0,+-n,0), (0,0,0,+-n), and is divided by 8 sets of parallel hyperplanes x_1+-x_2+-x_3+-x_4 = -(n-2),-(n-4),...,n-4,n-2. Then the unit complete 16-cells can be described to have centers in the following two groups:

(i) {(x_1,x_2,x_3,x_4) : x_i in Z, Sum |x_i| <= n-1, Sum x_i !== n (mod 2)};

(ii) {(x_1,x_2,x_3,x_4) : x_i in Z+1/2, Sum |x_i| <= n-1}.

The numbers of 16-cells in the two groups are respectively n^2*(n^2+2)/3 = A014820(n+1) and 2*(n+1)*n*(n-1)*(n-2)/3.

E.g.: for n = 2, a 2-scale 16-cell composed of a 16-cell honeycomb has 8 complete 16-cells centered at the permutations of (+-1,0,0,0), and 16 halved 16-cells centered at (+-1/2,+-1/2,+-1/2,+-1/2).

For n = 3, a 3-scale 16-cell composed of a 16-cell honeycomb has 49 complete 16-cells centered at (0,0,0,0), the permutations of (+-2,0,0,0), the permutations of (+-1,+-1,0,0), or (+-1/2,+-1/2,+-1/2,+-1/2), and 64 halved 16-cells centered at the permutations of (+-3/2,+-1/2,+-1/2,+-1/2).

Links

Jianing Song, Table of n, a(n) for n = 0..10000

Wikipedia, 16-cell honeycomb

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = n*(3*n^3-4*n^2+4)/3.

G.f.: x*(1+3*x+19*x^2+x^3)/(1-x)^5.

E.g.f.: x*(3+9*x+14*x^2+3*x^3)*exp(x)/3.

Example

For n = 3, among the 49 complete 16-cells,
  - 1 has vertices (0,0,0,0) + {(+-1,0,0,0),(0,+-1,0,0),(0,0,+-1,0),(0,0,0,+-1)};
  - 8 have vertices (+-2,0,0,0) and the permutations + {(+-1,0,0,0),(0,+-1,0,0),(0,0,+-1,0),(0,0,0,+-1)};
  - 24 have vertices (+-1,+-1,0,0) and the permutations + {(+-1,0,0,0),(0,+-1,0,0),(0,0,+-1,0),(0,0,0,+-1)};
  - 16 have vertices {(x_1,0,0,0),(0,x_2,0,0),(0,0,x_3,0),(0,0,0,x_4),(0,x_2,x_3,x_4),(x_1,0,x_3,x_4),(x_1,x_2,0,x_4),(x_1,x_2,x_3,0)}, where each x_i can be chosen from +1 or -1.

Mathematica

a[n_]:=n^4-4(n^3-n)/3; Array[a, 41, 0] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 8, 49, 176}, 41] (* or *) CoefficientList[Series[x*(1+3*x+19*x^2+x^3)/(1-x)^5, {x, 0, 40}], x] (* James C. McMahon, Oct 27 2025 *)

Prog

(PARI) a(n) = n^4 - 4*(n^3-n)/3

Crossrefs

Cf. A102860, A014820.

Sequence in context: A306049 A317226 A329051 * A295777 A319959 A270007

Adjacent sequences: A389087 A389088 A389089 * A389091 A389092 A389093

Keyword

nonn,easy

Author

Jianing Song, Oct 23 2025

Status

approved