OFFSET
0,3
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).
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
MAPLE
MATHEMATICA
Table[Floor[(3 n + 4)/4], {n, 0, 75}]
PROG
(PARI) a(n)=(3*n+4)\4 \\ Charles R Greathouse IV, Apr 16 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, Jan 06 2002
STATUS
approved