A006946 Independence number of De Bruijn graph of order n on two symbols. (Formerly M0834)

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

Offset

1,2

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

Table of n, a(n) for n=1..13.

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

Mathematica

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

Prog

(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)

Crossrefs

Cf. A000031, A333077, A333078.

Sequence in context: A088172 A048573 A221834 * A074129 A233042 A055003

Adjacent sequences: A006943 A006944 A006945 * A006947 A006948 A006949

Keyword

nonn,more,hard

Author

N. J. A. Sloane, Herb Taylor

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

Status

approved