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A006946
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Independence number of De Bruijn graph of order n on two symbols.
(Formerly M0834)
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5
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1, 2, 3, 7, 13, 28, 55, 114, 227, 466, 931, 1891, 3781
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OFFSET
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1,2
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COMMENTS
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Proposition 4.3 (b) in Lichiardopol's paper (see links) can be formulated as a(n) <= 2^(n-1) - A000031(n)/2 + 1 for odd n. For even n, Proposition 5.4 says that a(n) <= (a(n+1) + 1)/2 <= 2^(n-1) - A000031(n+1)/4 + 1. For n<=13, equality holds in both cases, and I conjecture that it holds for all n. If this is true, the sequence would continue a(14)=7645, a(15)=15289, a(16)=30841, a(17)=61681, ... - Pontus von Brömssen, Feb 29 2020
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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MATHEMATICA
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Length /@ Table[FindIndependentVertexSet[DeBruijnGraph[2, n]][[1]], {n, 6}]
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PROG
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(Python)
import networkx as nx
def deBruijn(n):
return nx.MultiDiGraph(((0, 0), (0, 0))) if n==0 else nx.line_graph(deBruijn(n-1))
return nx.max_weight_clique(nx.complement(nx.Graph(deBruijn(n))), weight=None)[1] #Pontus von Brömssen, Mar 07 2020 (updated Nov 12 2023)
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CROSSREFS
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KEYWORD
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nonn,more,hard
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AUTHOR
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EXTENSIONS
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a(7) from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 07 2010
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STATUS
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approved
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