1,2

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

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

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

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)

Cf. A000031, A333077, A333078.

Sequence in context: A088172 A048573 A221834 * A074129 A233042 A175248

Adjacent sequences: A006943 A006944 A006945 * A006947 A006948 A006949

nonn,more,hard

N. J. A. Sloane, Herb Taylor

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

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

approved