A033439 Number of edges in 7-partite Turán graph of order n.

0, 0, 1, 3, 6, 10, 15, 21, 27, 34, 42, 51, 61, 72, 84, 96, 109, 123, 138, 154, 171, 189, 207, 226, 246, 267, 289, 312, 336, 360, 385, 411, 438, 466, 495, 525, 555, 586, 618, 651, 685, 720, 756, 792, 829, 867, 906, 946, 987, 1029, 1071, 1114, 1158, 1203, 1249

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apart from the initial term this is the elliptic troublemaker sequence R_n(1,7) (also sequence R_n(6,7)) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below. - Peter Bala, Aug 12 2013

References

• Graham et al., Handbook of Combinatorics, Vol. 2, p. 1234.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• K. E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials, arXiv:1108.3051 [math.NT], 2011-2014.
• Eric Weisstein's World of Mathematics, Turán Graph [_Reinhard Zumkeller_, Nov 30 2009]
• Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Crossrefs

Cf. A002620, A000212, A033436, A033437, A033438, A033440, A033441, A033442, A033443, A033444. [From Reinhard Zumkeller, Nov 30 2009]

Elliptic troublemaker sequences: A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6)), A184535 (= R_n(2,5) = R_n(3,5)).

Programs

Magma

[Floor(3*n^2/7): n in [0..60]]; // Vincenzo Librandi, Oct 19 2013

Mathematica

CoefficientList[Series[- x^2 (x + 1) (x^2 - x + 1) (x^2 + x + 1)/((x - 1)^3 (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 19 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,1,-2,1},{0,0,1,3,6,10,15,21,27},60] (* Harvey P. Dale, Mar 19 2015 *)

Formula

a(n) = Sum_{k=0..n} A109720(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]

G.f.: -x^2*(x+1)*(x^2-x+1)*(x^2+x+1)/((x-1)^3*(x^6+x^5+x^4+x^3+x^2+x+1)). [Colin Barker, Aug 09 2012]

a(n) = floor(3*n^2/7). - Peter Bala, Aug 12 2013

a(0)=0, a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10, a(6)=15, a(7)=21, a(8)=27, a(n)=2*a(n-1)-a(n-2)+a(n-7)-2*a(n-8)+a(n-9). - Harvey P. Dale, Mar 19 2015

a(n) = Sum_{i=1..n} floor(6*i/7). - Wesley Ivan Hurt, Sep 12 2017

Extensions

More terms from Vincenzo Librandi, Oct 19 2013