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%I #42 Jan 31 2023 08:30:59
%S 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,
%T 21,22,22,23,24,25,25,26,27,28,28,29,30,31,31,32,33,34,34,35,36,37,37,
%U 38,39,40,40,41,42,43,43,44,45,46,46,47,48,49,49,50,51,52,52,53,54,55
%N a(n) = floor((3*n + 4)/4).
%C From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010: (Start)
%C 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.
%C A nice open-access proof that a(n-1) disjoint edge covers exist is given in Alon et al. (2009), who rediscovered the result.
%C 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)
%C 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
%H Noga Alon et al., <a href="https://doi.org/10.1007/s00454-009-9171-5">Polychromatic Colorings of Plane Graphs</a>, Discrete and Computational Geometry 42 (2009), 421-442. [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]
%H Lars Døvling Andersen, <a href="https://doi.org/10.1016/0012-365X(79)90076-1">Lower bounds on the cover-index of a graph</a>, Discrete Mathematics 25 (1979), 199-210. [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]
%H Ram P. Gupta, <a href="https://doi.org/10.1007/BFb0070378">On the chromatic index and the cover index of a multigraph</a>, Lecture Notes in Mathematics Volume 642, Springer, 1978, pages 204-215. [From David Pritchard (daveagp(AT)alum.mit.edu), May 07 2010]
%H John A. Pelesko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pelesko/pel11.html">Generalizing the Conway-Hofstadter $10,000 Sequence</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F G.f.: (1 + x^2 + x^3)/((1 - x)*(1 - x^4)).
%F a(n) = 1 + floor(3*n/4).
%F a(n) = (1/8)*(6*n + 5 + (-1)^n - 2*(-1)^floor((n-1)/2)). - _Ralf Stephan_, Jun 10 2005
%F Sum_{n>=0} (-1)^n/a(n) = log(3)/2 - Pi/(6*sqrt(3)). - _Amiram Eldar_, Jan 31 2023
%p A037915:=n->floor((3*n + 4)/4); seq(A037915(n), n=0..100); # _Wesley Ivan Hurt_, Nov 30 2013
%t Table[Floor[(3 n + 4)/4], {n, 0, 75}]
%o (PARI) a(n)=(3*n+4)\4 \\ _Charles R Greathouse IV_, Apr 16 2012
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_.
%E More terms from _Robert G. Wilson v_, Jan 06 2002
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