A033436 a(n) = ceiling( (3*n^2 - 4)/8 ).

0, 0, 1, 3, 6, 9, 13, 18, 24, 30, 37, 45, 54, 63, 73, 84, 96, 108, 121, 135, 150, 165, 181, 198, 216, 234, 253, 273, 294, 315, 337, 360, 384, 408, 433, 459, 486, 513, 541, 570, 600, 630, 661, 693, 726, 759, 793, 828

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of edges in 4-partite Turan graph of order n.

Apart from the initial term this equals the elliptic troublemaker sequence R_n(1,4) (also sequence R_n(3,4)) 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 08 2013

References

• R. L. Graham, Martin Grötschel, and László Lovász, Handbook of Combinatorics, Vol. 2, 1995, p. 1234.

Links

• Ivan Panchenko, 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.
• Katherine 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,-1,0,1,-2,1).

Crossrefs

Cf. A002620 (= R_n(1,2)), A000212 (= R_n(1,3) = R_n(2,3)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A033439 (= R_n(1,7) = R_n(6,7)), A033440, A033441, A033442, A033443, A033444.

Cf. A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6)), A184535 (= R_n(2,5) = R_n(3,5)).

Programs

Mathematica

LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 0, 1, 3, 6, 9}, 48] (* Jean-François Alcover, Sep 21 2017 *)

PARI

a(n)=(3*n^2 +3)\8 \\ Charles R Greathouse IV, Sep 24 2015

Formula

The second differences of the listed terms are periodic with period (1, 1, 1, 0) of length 4, showing that the terms satisfy the recurrence a(n) = 2a(n-1)-a(n-2)+a(n-4)-2a(n-5)+a(n-6). - John W. Layman, Jan 23 2001

a(n) = (1/16) {6n^2 - 5 + (-1)^n + 2(-1)^[n/2] - 2(-1)^[(n-1)/2] }. Therefore a(n) is asymptotic to 3/8*n^2. - Ralf Stephan, Jun 09 2005

O.g.f.: -x^2*(1+x+x^2)/((x+1)*(x^2+1)*(x-1)^3). - R. J. Mathar, Dec 05 2007

a(n) = Sum_{k=0..n} A166486(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009

a(n) = floor(3*n^2/8). - Peter Bala, Aug 08 2013

a(n) = Sum_{i=1..n} floor(3*i/4). - Wesley Ivan Hurt, Sep 12 2017

Sum_{n>=2} 1/a(n) = Pi^2/36 + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)) + 2/3. - Amiram Eldar, Sep 24 2022