

A037915


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


8



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

a(n1) 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(n1) disjoint edge covers (sets of edges touching every vertex), and this is tight. [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]
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.


REFERENCES

R. Gupta, On the chromatic index and the cover index of a multigraph, Lecture Notes in Mathematics Volume 642, Springer, 1978, pages 204215. doi:10.1007/BFb0070378 [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]
N. Alon et al., Polychromatic Colorings of Plane Graphs, Discrete and Computational Geometry 42 (2009), 421442. doi:10.1007/s0045400991715 [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]
L. Andersen, Lower bounds on the coverindex of a graph, Discrete Mathematics 25 (1979), 199210. doi:10.1016/0012365X(79)900761 [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]


LINKS

Table of n, a(n) for n=0..72.
John A. Pelesko, Generalizing the ConwayHofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
A nice openaccess proof that a(n1) disjoint edge covers exist is given in Alon et al. (2009), who rediscovered the result. [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]
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((n1)/2)).  Ralf Stephan, Jun 10 2005


EXAMPLE

E.g. every multigraph with minimum degree 7 contains a(71)=5 disjoint edge covers. This is tight for a 3vertex 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. [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]


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}]


PROG

(PARI) a(n)=(3*n+1)\4 \\ Charles R Greathouse IV, Apr 16 2012


CROSSREFS

Sequence in context: A123580 A072894 A328309 * A195180 A069210 A195172
Adjacent sequences: A037912 A037913 A037914 * A037916 A037917 A037918


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane.


EXTENSIONS

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


STATUS

approved



