A111384 a(n) = binomial(n,3) - binomial(floor(n/2),3) - binomial(ceiling(n/2),3).

0, 0, 0, 1, 4, 9, 18, 30, 48, 70, 100, 135, 180, 231, 294, 364, 448, 540, 648, 765, 900, 1045, 1210, 1386, 1584, 1794, 2028, 2275, 2548, 2835, 3150, 3480, 3840, 4216, 4624, 5049, 5508, 5985, 6498, 7030, 7600, 8190, 8820, 9471, 10164, 10879, 11638, 12420, 13248

Offset: 0

Author

N. J. A. Sloane, Nov 10 2005

Keywords

• nonn
• easy

Comments

a(n) is the maximum number of open triangles in a simple, undirected graph with n vertices. - Eugene Lykhovyd, Oct 20 2018

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

a(n) is the maximum number of triples (i.e., 3-element subsets of {1..n}) such that there exists a 2-coloring of {1..n} in which no triple is monochromatic. For the contrasting minimum number of triples such that every 2-coloring of {1..n} results in at least one monochromatic triple, see A385403. - David Dewan, Jul 04 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Peter Keevash and Benny Sudakov, The Turán number of the Fano plane, Combinatorica, 25 (2005), 561-574; alternative link.
• Artem Pyatkin, Eugene Lykhovyd, and Sergiy Butenko, The maximum number of induced open triangles in graphs of a given order, Optimization Letters 13, 1927-1935 (2019).
• Adityanarayanan Radhakrishnan, Liam Solus, and Caroline Uhler, Counting Markov Equivalence Classes by Number of Immoralities, arXiv preprint arXiv:1611.07493 [math.CO], 2016-2017.
• Markus Sigg, A note on OEIS sequence A111384, arXiv:2309.14840 [math.CO], 2023.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A006578, A006918.

Programs

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

Magma

[Binomial(n, 3) - Binomial(Floor(n/2), 3) - Binomial(Ceiling(n/2), 3): n in [0..50]]; // Vincenzo Librandi, Oct 20 2018

Maple

seq(floor(n/2)*ceil(n/2)*(n-2)/2,n=0..50); # James R. Buddenhagen, Nov 11 2009

Mathematica

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 0, 0, 1, 4, 9}, 50] (* Vincenzo Librandi, Oct 20 2018 *)

PARI

a(n)=floor(n/2)*ceil(n/2)*(n-2)/2 \\ Charles R Greathouse IV, Oct 16 2015

Formula

a(n) = floor(n/2)*ceiling(n/2)*(n-2)/2. - James R. Buddenhagen, Nov 11 2009

From R. J. Mathar, Mar 18 2010: (Start)

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).

G.f.: x^3*(1+2*x)/ ((1+x)^2 * (x-1)^4). (End)

a(n) = A006918(n-2) + 2*A006918(n-3). - R. J. Mathar, Jan 20 2018

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

Sum_{n>=3} 1/a(n) = 4/3 - Pi^2/6 + 8*log(2)/3. - Amiram Eldar, Oct 10 2023

E.g.f.: (x + 2)*(x*(x - 1)*cosh(x) + (x^2 - x + 1)*sinh(x))/8. - Stefano Spezia, Apr 08 2024