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%I #61 Dec 14 2023 05:44:34
%S 0,0,1,3,6,9,13,18,24,30,37,45,54,63,73,84,96,108,121,135,150,165,181,
%T 198,216,234,253,273,294,315,337,360,384,408,433,459,486,513,541,570,
%U 600,630,661,693,726,759,793,828
%N a(n) = ceiling( (3*n^2 - 4)/8 ).
%C Number of edges in 4-partite Turan graph of order n.
%C 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
%D R. L. Graham, Martin Grötschel, and László Lovász, Handbook of Combinatorics, Vol. 2, 1995, p. 1234.
%H Ivan Panchenko, <a href="/A033436/b033436.txt">Table of n, a(n) for n = 0..10000</a>
%H Kevin Beanland, Hung Viet Chu, and Carrie E. Finch-Smith, <a href="https://arxiv.org/abs/2112.14905">Generalized Schreier sets, linear recurrence relation, Turán graphs</a>, arXiv:2112.14905 [math.CO], 2021.
%H Katherine 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>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tur%C3%A1n_graph">Turán graph</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F 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
%F 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
%F O.g.f.: -x^2*(1+x+x^2)/((x+1)*(x^2+1)*(x-1)^3). - _R. J. Mathar_, Dec 05 2007
%F a(n) = Sum_{k=0..n} A166486(k)*(n-k). - _Reinhard Zumkeller_, Nov 30 2009
%F a(n+1) = n + floor(n^2/3). - _Gary Detlefs_, Mar 27 2010
%F a(n) = floor(3*n^2/8). - _Peter Bala_, Aug 08 2013
%F a(n) = Sum_{i=1..n} floor(3*i/4). - _Wesley Ivan Hurt_, Sep 12 2017
%F 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
%t LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 0, 1, 3, 6, 9}, 48] (* _Jean-François Alcover_, Sep 21 2017 *)
%o (PARI) a(n)=(3*n^2 +3)\8 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y 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.
%Y Cf. 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_
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