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A033440 Number of edges in 8-partite Turán graph of order n.

%I A033440 #35 Feb 16 2025 08:32:36
%S A033440 0,0,1,3,6,10,15,21,28,35,43,52,62,73,85,98,112,126,141,157,174,192,
%T A033440 211,231,252,273,295,318,342,367,393,420,448,476,505,535,566,598,631,
%U A033440 665,700,735,771,808,846,885,925
%N A033440 Number of edges in 8-partite Turán graph of order n.
%D A033440 Graham et al., Handbook of Combinatorics, Vol. 2, p. 1234.
%H A033440 Vincenzo Librandi, <a href="/A033440/b033440.txt">Table of n, a(n) for n = 0..1000</a>
%H A033440 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TuranGraph.html">Turán Graph</a> [_Reinhard Zumkeller_, Nov 30 2009]
%H A033440 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tur%C3%A1n_graph">Turán graph</a> [_Reinhard Zumkeller_, Nov 30 2009]
%H A033440 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,1,-2,1).
%F A033440 a(n) = round( (7/16)*n(n-2) ) +0 or -1 depending on n: if there is k such 8k+4<=n<=8k+6 then a(n) = floor( (7/16)*n*(n-2)) otherwise a(n) = round( (7/16)*n(n-2)). E.g. because 8*2+4<=21<=8*2+6 a(n) = floor((7/16)*21*19) = floor(174, 5625)=174. - _Benoit Cloitre_, Jan 17 2002
%F A033440 a(n) = Sum_{k=0..n} A168181(k)*(n-k). [_Reinhard Zumkeller_, Nov 30 2009]
%F A033440 G.f.: -x^2*(x^6+x^5+x^4+x^3+x^2+x+1)/((x-1)^3*(x+1)*(x^2+1)*(x^4+1)). [_Colin Barker_, Aug 09 2012]
%F A033440 a(n) = Sum_{i=1..n} floor(7*i/8). - _Wesley Ivan Hurt_, Sep 12 2017
%t A033440 CoefficientList[Series[- x^2 (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)/((x - 1)^3 (x + 1) (x^2 + 1) (x^4 + 1)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 19 2013 *)
%t A033440 LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1},{0,0,1,3,6,10,15,21,28,35},50] (* _Harvey P. Dale_, Mar 23 2015 *)
%Y A033440 Cf. A002620, A000212, A033436, A033437, A033438, A033439, A033441, A033442, A033443, A033444. [_Reinhard Zumkeller_, Nov 30 2009]
%K A033440 nonn,easy
%O A033440 0,4
%A A033440 _N. J. A. Sloane_