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A037915
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a(n) = floor((3*n + 4)/4).
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8
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MAPLE
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MATHEMATICA
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Table[Floor[(3 n + 4)/4], {n, 0, 75}]
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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