A130809 If X_1, ..., X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 3-subsets of X containing none of X_i, (i=1,...,n).

8, 32, 80, 160, 280, 448, 672, 960, 1320, 1760, 2288, 2912, 3640, 4480, 5440, 6528, 7752, 9120, 10640, 12320, 14168, 16192, 18400, 20800, 23400, 26208, 29232, 32480, 35960, 39680, 43648, 47872, 52360, 57120, 62160, 67488, 73112, 79040, 85280

Offset: 3

Author

Milan Janjic, Jul 16 2007

Keywords

• nonn
• easy

Comments

Uncentered octahedral numbers: take a simple cubical grid of size n X n X n where n = 2k is an even number, n >= 6. Retain all points that are at Manhattan distance n or greater from all 8 corners of the cube, and discard all other points. The number of points that remain is a(k). If n were to be an odd number, the same operation would yield the centered octahedral numbers A001845. - Arun Giridhar, Mar 06 2014

For an (n+2)-dimensional Rubik's cube, the number of cubes that have exactly 3 exposed facets. - Phil Scovis, Aug 03 2009

a(n) is the number of 2-simplices in an n-cross polytope. - Arkadiusz Wesolowski, Oct 16 2012

a(n) is also the number of unit tetrahedra in an (n+1)-scaled octahedron composed of the tetrahedral-octahedral honeycomb. - Jason Pruski, Aug 31 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 3..1000
• Harlan J. Brothers, Pascal's Prism: Supplementary Material, 2012.
• Milan Janjic, Two Enumerative Functions
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000079, A001787, A001788, A001789, A001845, A002409, A003472, A038207, A046092, A054849, A054851, A056220, A140325, A140354.

Programs

Magma

[(4/3)*n*(n-1)*(n-2): n in [3..60]]; // Vincenzo Librandi, Oct 03 2017

Maple

a:=n->4/3*n*(n-1)*(n-2);

Mathematica

Table[(4/3) n (n - 1) (n - 2), {n, 3, 41}] (* or *)
Table[Binomial[n, n - 3] 2^3, {n, 3, 41}] (* or *)
DeleteCases[#, 0] &@ CoefficientList[Series[8 x^3/(1 - x)^4, {x, 0, 41}], x] (* Michael De Vlieger, Aug 31 2017 *)

PARI

a(n) = 4*n*(n-1)*(n-2)/3; \\ Andrew Howroyd, Nov 06 2018

Formula

a(n) = (4/3)*n*(n-1)*(n-2).

a(n) = C(n,n-3)*8, n >= 3. - Zerinvary Lajos, Dec 07 2007

G.f.: 8*x^3/(1-x)^4. - Colin Barker, Apr 14 2012

For n>1, a(n) = a(n-1) + A056220(n-1) + A056220(n-2). - Bruce J. Nicholson, Feb 14 2018

From Amiram Eldar, Mar 24 2022: (Start)

Sum_{n>=3} 1/a(n) = 3/16.

Sum_{n>=3} (-1)^(n+1)/a(n) = 3*log(2)/2 - 15/16. (End)

E.g.f.: 4*x^3*exp(x)/3. - Stefano Spezia, Apr 02 2024