OFFSET
0,5
COMMENTS
a(n+1) = floor((n^3-n)/12) is an upper bound for the Kirchhoff index of a circulant graph with n vertices [Zhang & Yang]. - R. J. Mathar, Apr 26 2007
Also the matching number of the n-tetrahedral graph. - Eric W. Weisstein, Jun 20 2017
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Johnson Graph
Eric Weisstein's World of Mathematics, Matching Number
Eric Weisstein's World of Mathematics, Tetrahedral Graph
H. Zhang and Y. Yang, Resistance Distance and Kirchhoff Index in Circulant Graphs, Int. J. Quant. Chem. 107 (2007) 330-339.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-3,3,-1).
FORMULA
From R. J. Mathar, Apr 15 2010: (Start)
a(n) = +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7).
G.f.: x^4*(2-x+x^2) / ( (1-x)^4*(1+x)*(1+x^2) ). (End)
a(n) = (1/24)*(2*n^3 - 6*n^2 + 4*n - 3*(1-(-1)^n)*(1 - (-1)^((2*n-1+(-1)^n)/4)) ). - Luce ETIENNE, Jun 26 2014
MAPLE
seq(floor(binomial(n, 3)/2), n=0..40); # Zerinvary Lajos, Jan 12 2009
MATHEMATICA
CoefficientList[Series[x^4*(2-x+x^2)/((1-x)^3*(1-x^4)), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 07 2012 *)
(* Contributions from Eric W. Weisstein, Jun 20 2017 *)
Table[(3*((-1)^n -1) + 2*n*(n-1)*(n-2) + 6*Sin[(n*Pi)/2])/24, {n, 0, 50}]
LinearRecurrence[{3, -3, 1, 1, -3, 3, -1}, {0, 0, 0, 2, 5, 10, 17}, 50] (* End *)
Floor[Binomial[Range[0, 50], 3]/2] (* G. C. Greubel, Oct 06 2024 *)
PROG
(Sage) [floor(binomial(n, 3)/2) for n in range(41)] # Zerinvary Lajos, Dec 01 2009
(Magma) [Floor(n*(n-1)*(n-2)/12): n in [0..50]]; // Vincenzo Librandi, Jul 07 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved