

A130809


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


15



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
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OFFSET

3,1


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 2simplices in an ncross polytope.  Arkadiusz Wesolowski, Oct 16 2012
a(n) is also the number of unit tetrahedra in an (n+1)scaled octahedron composed of the tetrahedraloctahedral honeycomb.  Jason Pruski, Aug 31 2017


LINKS

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


FORMULA

a(n) = (4/3)*n*(n1)*(n2).
a(n) = C(n,n3)*8, n >= 3.  Zerinvary Lajos, Dec 07 2007
G.f.: 8*x^3/(1x)^4.  Colin Barker, Apr 14 2012
For n>1, a(n) = a(n1) + A056220(n1) + A056220(n2).  Bruce J. Nicholson, Feb 14 2018


MAPLE

a:=n>4/3*n*(n1)*(n2);


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 *)


PROG

(MAGMA) [(4/3)*n*(n1)*(n2): n in [3..60]]; // Vincenzo Librandi, Oct 03 2017
(PARI) a(n) = 4*n*(n1)*(n2)/3; \\ Andrew Howroyd, Nov 06 2018


CROSSREFS

Cf. A000079, A001787, A001788, A001789, A002409, A003472, A038207, A046092, A054849, A054851, A056220, A140325, A140354.
Sequence in context: A139098 A224543 A211633 * A333174 A018839 A008412
Adjacent sequences: A130806 A130807 A130808 * A130810 A130811 A130812


KEYWORD

nonn,easy


AUTHOR

Milan Janjic, Jul 16 2007


STATUS

approved



