A006946 Independence number of de Bruijn graph of order n on two symbols.

1, 2, 3, 7, 13, 28, 55, 114, 227, 466, 931, 1891, 3781

Offset: 1

Author

N. J. A. Sloane, Herb Taylor

Keywords

• nonn
• more
• hard

Comments

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

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Pontus von Brömssen, Maximum independent sets for de Bruijn graphs of order 8 to 13
• N. Lichiardopol, Independence number of de Bruijn graphs, Discrete Math., 306 (2006), no.12, 1145-1160. [Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 07 2010]
• Eric Weisstein's World of Mathematics, de Bruijn Graph
• Eric Weisstein's World of Mathematics, Independence Number

Crossrefs

Cf. A000031, A333077, A333078.

Programs

Mathematica

Length /@ Table[FindIndependentVertexSet[DeBruijnGraph[2, n]][[1]], {n, 6}]

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))
def A006946(n):
    return nx.max_weight_clique(nx.complement(nx.Graph(deBruijn(n))),weight=None)[1] #Pontus von Brömssen, Mar 07 2020 (updated Nov 12 2023)

Extensions

a(7) from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 07 2010

a(8) to a(13) from Pontus von Brömssen, Feb 29 2020