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

0,4

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).

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+1) = n + floor(n^2/3). - Gary Detlefs, Mar 27 2010

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

Mathematica

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

Prog

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

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)).

Sequence in context: A048202 A014785 A132352 * A059293 A002578 A129728

Adjacent sequences: A033433 A033434 A033435 * A033437 A033438 A033439

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved