A033437 Number of edges in 5-partite Turán graph of order n.

0, 0, 1, 3, 6, 10, 14, 19, 25, 32, 40, 48, 57, 67, 78, 90, 102, 115, 129, 144, 160, 176, 193, 211, 230, 250, 270, 291, 313, 336, 360, 384, 409, 435, 462, 490, 518, 547, 577, 608, 640, 672, 705, 739, 774, 810, 846, 883, 921, 960, 1000, 1040, 1081, 1123, 1166, 1210, 1254

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apart from the initial term this is the elliptic troublemaker sequence R_n(1,5) (also sequence R_n(4,5)) 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

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

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Kevin Beanland, Hung Viet Chu, and Carrie E. Finch-Smith, Generalized Schreier sets, linear recurrence relation, Turán graphs, arXiv:2112.14905 [math.CO], 2021.
• 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
• Wikipedia, Turán graph
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Crossrefs

Cf. A002620, A000212, A033436, A033438, A033439, A033440, A033441, A033442, A033443, A033444. - 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)).

Cf. A279169.

Programs

Magma

[2*n^2 div 5: n in [0..60]]; // Vincenzo Librandi, Apr 20 2015

Mathematica

Table[Floor[2n^2/5],{n,0,60}]

PARI

a(n)=2*n^2\5 \\ Charles R Greathouse IV, Apr 20 2015

Formula

G.f.: (x^5+x^4+x^3+x^2)/((1-x^5)*(1-x)^2).

a(n) = Sum_{k=0..n} A011558(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009

a(n) = floor( 2n^2/5 ). - Wesley Ivan Hurt, Jun 20 2013

a(n) = Sum_{i=1..n} floor(4*i/5). - Wesley Ivan Hurt, Sep 12 2017