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%I #75 Apr 08 2024 18:47:42
%S 0,0,0,1,4,9,18,30,48,70,100,135,180,231,294,364,448,540,648,765,900,
%T 1045,1210,1386,1584,1794,2028,2275,2548,2835,3150,3480,3840,4216,
%U 4624,5049,5508,5985,6498,7030,7600,8190,8820,9471,10164,10879,11638,12420,13248
%N a(n) = binomial(n,3) - binomial(floor(n/2),3) - binomial(ceiling(n/2),3).
%C a(n) is the maximum number of open triangles in a simple, undirected graph with n vertices. - _Eugene Lykhovyd_, Oct 20 2018
%C a(n) is the maximum number of elements of the set T := {3} u (IN \ 3IN) that can be written as a sum of three distinct elements of an n-element subset of T, see arXiv link 2309.14840. - _Markus Sigg_, Sep 27 2023
%H Vincenzo Librandi, <a href="/A111384/b111384.txt">Table of n, a(n) for n = 0..10000</a>
%H Peter Keevash and Benny Sudakov, <a href="https://doi.org/10.1007/s00493-005-0034-2">The Turan number of the Fano plane</a>, Combinatorica, 25 (2005), 561-574; <a href="http://people.maths.ox.ac.uk/keevash/papers/fano-journal.pdf">alternative link</a>.
%H Artem Pyatkin, Eugene Lykhovyd, and Sergiy Butenko, <a href="https://doi.org/10.1007/s11590-018-1330-2">The maximum number of induced open triangles in graphs of a given order</a>, Optimization Letters (2018).
%H Adityanarayanan Radhakrishnan, Liam Solus, and Caroline Uhler, <a href="http://arxiv.org/abs/1611.07493">Counting Markov Equivalence Classes by Number of Immoralities</a>, arXiv preprint arXiv:1611.07493 [math.CO], 2016-2017.
%H Markus Sigg, <a href="https://arxiv.org/abs/2309.14840">A note on OEIS sequence A111384</a>, arXiv:2309.14840 [math.CO], 2023.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F a(n) = floor(n/2)*ceiling(n/2)*(n-2)/2. - _James R. Buddenhagen_, Nov 11 2009
%F From _R. J. Mathar_, Mar 18 2010: (Start)
%F a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
%F G.f.: x^3*(1+2*x)/ ((1+x)^2 * (x-1)^4). (End)
%F a(n) = A006918(n-2) + 2*A006918(n-3). - _R. J. Mathar_, Jan 20 2018
%F a(n) = (n-2)*n^2/8 for even n, a(n) = (n-2)*(n^2-1)/8 for odd n. - _Markus Sigg_, Sep 26 2023
%F Sum_{n>=3} 1/a(n) = 4/3 - Pi^2/6 + 8*log(2)/3. - _Amiram Eldar_, Oct 10 2023
%F E.g.f.: (x + 2)*(x*(x - 1)*cosh(x) + (x^2 - x + 1)*sinh(x))/8. - _Stefano Spezia_, Apr 08 2024
%p seq(floor(n/2)*ceil(n/2)*(n-2)/2,n=0..50); # _James R. Buddenhagen_, Nov 11 2009
%t LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 0, 0, 1, 4, 9}, 50] (* _Vincenzo Librandi_, Oct 20 2018 *)
%o (PARI) a(n)=floor(n/2)*ceil(n/2)*(n-2)/2 \\ _Charles R Greathouse IV_, Oct 16 2015
%o (Magma) [Binomial(n, 3) - Binomial(Floor(n/2), 3) - Binomial(Ceiling(n/2), 3): n in [0..50]]; // _Vincenzo Librandi_, Oct 20 2018
%o (GAP) a:=[0,0,0,1,4,9];; for n in [7..50] do a[n]:=2*a[n-1]+a[n-2]-4*a[n-3]+a[n-4]+2*a[n-5]-a[n-6]; od; a; # _Muniru A Asiru_, Oct 21 2018
%Y Cf. A006578, A006918.
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_, Nov 10 2005
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