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%I #39 Sep 08 2022 08:44:51
%S 0,0,1,3,6,10,15,21,27,34,42,51,61,72,84,96,109,123,138,154,171,189,
%T 207,226,246,267,289,312,336,360,385,411,438,466,495,525,555,586,618,
%U 651,685,720,756,792,829,867,906,946,987,1029,1071,1114,1158,1203,1249
%N Number of edges in 7-partite Turán graph of order n.
%C 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
%D Graham et al., Handbook of Combinatorics, Vol. 2, p. 1234.
%H Vincenzo Librandi, <a href="/A033439/b033439.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TuranGraph.html">Turán Graph</a> [_Reinhard Zumkeller_, Nov 30 2009]
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tur%C3%A1n_graph">Turán graph</a> [_Reinhard Zumkeller_, Nov 30 2009]
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,1,-2,1).
%F a(n) = Sum_{k=0..n} A109720(k)*(n-k). [_Reinhard Zumkeller_, Nov 30 2009]
%F 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]
%F a(n) = floor(3*n^2/7). - _Peter Bala_, Aug 12 2013
%F 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
%F a(n) = Sum_{i=1..n} floor(6*i/7). - _Wesley Ivan Hurt_, Sep 12 2017
%t 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 *)
%t 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 *)
%o (Magma) [Floor(3*n^2/7): n in [0..60]]; // _Vincenzo Librandi_, Oct 19 2013
%Y Cf. A002620, A000212, A033436, A033437, A033438, A033440, A033441, A033442, A033443, A033444. [From _Reinhard Zumkeller_, Nov 30 2009]
%Y 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 nonn,easy
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _Vincenzo Librandi_, Oct 19 2013
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