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

0,4

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).

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

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 *)

Prog

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

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)).

Sequence in context: A310081 A357779 A240443 * A194082 A061786 A171971

Adjacent sequences: A033436 A033437 A033438 * A033440 A033441 A033442

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Vincenzo Librandi, Oct 19 2013

Status

approved