A037915 a(n) = floor((3*n + 4)/4).

1, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 43, 43, 44, 45, 46, 46, 47, 48, 49, 49, 50, 51, 52, 52, 53, 54, 55

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010: (Start)

a(n-1) is the "cover index" guaranteed by a multigraph with minimum degree n. I.e., in a multigraph where every node has degree >=n, it contains a(n-1) disjoint edge covers (sets of edges touching every vertex), and this is tight.

A nice open-access proof that a(n-1) disjoint edge covers exist is given in Alon et al. (2009), who rediscovered the result.

E.g. every multigraph with minimum degree 7 contains a(7-1)=5 disjoint edge covers. This is tight for a 3-vertex graph: e.g. the multigraph with V = {a, b, c} and E = {4*ab, 4*bc, 3*ac} has minimum degree 7 does not have >5 disjoint edge covers.(End)

It appears that a (n) = number of distinct values among Floor(i^2 / n) for i = 0, 1, 2, ..., n. - Samuel Vodovoz, Jun 15 2015

Links

• Noga Alon et al., Polychromatic Colorings of Plane Graphs, Discrete and Computational Geometry 42 (2009), 421-442. [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]
• Lars Døvling Andersen, Lower bounds on the cover-index of a graph, Discrete Mathematics 25 (1979), 199-210. [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]
• Ram P. Gupta, On the chromatic index and the cover index of a multigraph, Lecture Notes in Mathematics Volume 642, Springer, 1978, pages 204-215. [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]
• John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Programs

Maple

A037915:=n->floor((3*n + 4)/4); seq(A037915(n), n=0..100); # Wesley Ivan Hurt, Nov 30 2013

Mathematica

Table[Floor[(3 n + 4)/4], {n, 0, 75}]

PARI

a(n)=(3*n+4)\4 \\ Charles R Greathouse IV, Apr 16 2012

Formula

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

a(n) = 1 + floor(3*n/4).

a(n) = (1/8)*(6*n + 5 + (-1)^n - 2*(-1)^floor((n-1)/2)). - Ralf Stephan, Jun 10 2005

Sum_{n>=0} (-1)^n/a(n) = log(3)/2 - Pi/(6*sqrt(3)). - Amiram Eldar, Jan 31 2023

Extensions

More terms from Robert G. Wilson v, Jan 06 2002