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A111384 a(n) = binomial(n,3) - binomial(floor(n/2),3) - binomial(ceiling(n/2),3). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

From Gary W. Adamson, Mar 03 2010: (Start)

Starting with 1 = M * [1, 2, 3, ...] where M = a matrix with (1, 4, 7, 10, ...)

in every column, shifted down twice for columns > 1. The row sums of triangle

M = A006578: (1, 4, 8, 14, 21, 30, 40, ...). (End)

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

P. Keevash et al., The Turan number of the Fano plane, Combinatorica, 25 (2005), 561-574.

Artem Pyatkin, Eugene Lykhovyd, Sergiy Butenko, The maximum number of induced open triangles in graphs of a given order, Optimization Letters (2018).

Adityanarayanan Radhakrishnan, Liam Solus, Caroline Uhler, Counting Markov Equivalence Classes by Number of Immoralities, arXiv preprint arXiv:1611.07493 [math.CO], 2016-2017.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

FORMULA

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

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 *)

PROG

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

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

(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

CROSSREFS

Cf. A006578, A006918.

Sequence in context: A008146 A038098 A299274 * A196039 A238091 A301017

Adjacent sequences:  A111381 A111382 A111383 * A111385 A111386 A111387

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 10 2005

STATUS

approved

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Last modified November 26 12:37 EST 2020. Contains 338639 sequences. (Running on oeis4.)