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A033438 Number of edges in 6-partite Turán graph of order n.

%I A033438 #36 Feb 16 2025 08:32:36
%S A033438 0,0,1,3,6,10,15,20,26,33,41,50,60,70,81,93,106,120,135,150,166,183,
%T A033438 201,220,240,260,281,303,326,350,375,400,426,453,481,510,540,570,601,
%U A033438 633,666,700,735,770,806,843,881
%N A033438 Number of edges in 6-partite Turán graph of order n.
%C A033438 Apart from the initial term this is the elliptic troublemaker sequence R_n(1,6) (also sequence R_n(5,6)) 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
%D A033438 Graham et al., Handbook of Combinatorics, Vol. 2, p. 1234.
%H A033438 K. E. Stange, <a href="https://arxiv.org/abs/1108.3051">Integral points on elliptic curves and explicit valuations of division polynomials</a>, arXiv:1108.3051 [math.NT], 2011-2014.
%H A033438 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TuranGraph.html">Turán Graph</a> [_Reinhard Zumkeller_, Nov 30 2009]
%H A033438 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tur%C3%A1n_graph">Turán graph</a> [_Reinhard Zumkeller_, Nov 30 2009]
%H A033438 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,1,-2,1).
%F A033438 a(n) = Sum_{k=0..n} A097325(k)*(n-k). - _Reinhard Zumkeller_, Nov 30 2009
%F A033438 a(n) = +2*a(n-1) -a(n-2) +a(n-6) -2*a(n-7) +a(n-8).
%F A033438 G.f.: -x^2*(1+x+x^3+x^4+x^2) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^3 ).
%F A033438 a(n) = floor(5*n^2/12). - _Peter Bala_, Aug 12 2013
%F A033438 a(n) = Sum_{i=1..n} floor(5*i/6). - _Wesley Ivan Hurt_, Sep 12 2017
%t A033438 a[n_] := Floor[5n^2/12];
%t A033438 Table[a[n], {n, 0, 46}] (* _Jean-François Alcover_, Jul 31 2018, after _Peter Bala_ *)
%Y A033438 Differs from A025708(n)+1 at 31st position.
%Y A033438 Cf. A002620, A000212, A033436, A033437, A033439, A033440, A033441, A033442, A033443, A033444. [From _Reinhard Zumkeller_, Nov 30 2009]
%Y A033438 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)).
%K A033438 nonn,easy
%O A033438 0,4
%A A033438 _N. J. A. Sloane_