A011894 - OEIS

%I #54 Oct 10 2024 05:23:51

%S 0,0,0,0,2,5,10,17,28,42,60,82,110,143,182,227,280,340,408,484,570,

%T 665,770,885,1012,1150,1300,1462,1638,1827,2030,2247,2480,2728,2992,

%U 3272,3570,3885,4218,4569,4940,5330,5740,6170,6622,7095,7590,8107,8648,9212,9800

%N a(n) = floor(n*(n-1)*(n-2)/12).

%C 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

%C Also the matching number of the n-tetrahedral graph. - _Eric W. Weisstein_, Jun 20 2017

%H Vincenzo Librandi, <a href="/A011894/b011894.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MatchingNumber.html">Matching Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TetrahedralGraph.html">Tetrahedral Graph</a>

%H H. Zhang and Y. Yang, <a href="http://dx.doi.org/10.1002/qua.21068">Resistance Distance and Kirchhoff Index in Circulant Graphs</a>, Int. J. Quant. Chem. 107 (2007) 330-339.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,1,-3,3,-1).

%F From _R. J. Mathar_, Apr 15 2010: (Start)

%F 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).

%F G.f.: x^4*(2-x+x^2) / ( (1-x)^4*(1+x)*(1+x^2) ). (End)

%F 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

%p seq(floor(binomial(n,3)/2), n=0..40); # _Zerinvary Lajos_, Jan 12 2009

%t CoefficientList[Series[x^4*(2-x+x^2)/((1-x)^3*(1-x^4)),{x, 0, 50}], x] (* _Vincenzo Librandi_, Jul 07 2012 *)

%t (* Contributions from _Eric W. Weisstein_, Jun 20 2017 *)

%t Table[(3*((-1)^n -1) + 2*n*(n-1)*(n-2) + 6*Sin[(n*Pi)/2])/24, {n,0,50}]

%t LinearRecurrence[{3,-3,1,1,-3,3,-1}, {0,0,0,2,5,10,17}, 50] (* End *)

%t Floor[Binomial[Range[0,50], 3]/2] (* _G. C. Greubel_, Oct 06 2024 *)

%o (Sage) [floor(binomial(n,3)/2) for n in range(41)] # _Zerinvary Lajos_, Dec 01 2009

%o (Magma) [Floor(n*(n-1)*(n-2)/12): n in [0..50]]; // _Vincenzo Librandi_, Jul 07 2012

%Y Cf. A011886.

%K nonn,easy

%O 0,5

%A _N. J. A. Sloane_