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 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).
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
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
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 *)
PROG
(Magma) [(4/3)*n*(n-1)*(n-2): n in [3..60]]; // Vincenzo Librandi, Oct 03 2017
(PARI) a(n) = 4*n*(n-1)*(n-2)/3; \\ Andrew Howroyd, Nov 06 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Milan Janjic, Jul 16 2007
STATUS
approved