A140462 Turan's upper bound on the number of triangles of a simplicial complex of dimension two for which every minimal non-face has three vertices.

0, 0, 0, 1, 3, 7, 14, 23, 36, 54, 75, 102, 136, 174, 220, 275, 335, 405, 486, 573, 672, 784, 903, 1036, 1184, 1340, 1512, 1701, 1899, 2115, 2350, 2595, 2860, 3146, 3443, 3762, 4104, 4458, 4836, 5239, 5655, 6097, 6566, 7049, 7560, 8100, 8655

Offset

0,5

Comments

Conjecture 1.2, p. 2 of Frohmader.

References

P. Turan, Research Problem, Kozl MTA Mat. Kutato Int. 6(1961)417-423.

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Andrew Frohmader, More Constructions for Turan's (3, 4)-Conjecture

Formula

a(n) = (5/2)*(k^3) - (3/2)*(k^2) if n = 3*k; (5/2)*(k^3) + (k^2) - (1/2)*k if n = 3*k+1; (5/2)*(k^3) + (7/2)*(k^2) + k if n = 3*k+2.

Empirical g.f.: x^3*(x^3+2*x^2+x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, May 04 2013

Maple

A140462 := proc(n) local k: k:=floor(n/3): if(n mod 3 = 0)then return 5*(k^3)/2 - 3*(k^2)/2: elif(n mod 3 = 1)then return 5*(k^3)/2 + k^2 - k/2: else return 5*(k^3)/2 + 7*(k^2)/2 + k: fi: end:
seq(A140462(n), n=0..40); # Nathaniel Johnston, Apr 26 2011

Mathematica

a[n_] := Which[k = Floor[n/3]; Mod[n, 3] == 0, 5*(k^3)/2 - 3*(k^2)/2, Mod[n, 3] == 1, 5*(k^3)/2 + k^2 - k/2, True, 5*(k^3)/2 + 7*(k^2)/2 + k];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 28 2017, from Maple *)

Crossrefs

Sequence in context: A294400 A115285 A004232 * A227841 A225256 A093523

Adjacent sequences: A140459 A140460 A140461 * A140463 A140464 A140465

Keyword

easy,nonn

Author

Jonathan Vos Post, Jun 27 2008

Status

approved