A166316 Lexicographically largest binary de Bruijn sequences, B(2,n).

2, 12, 232, 63056, 4221224224, 18295693635288736320, 338921575014037816709507133224870496384, 115563265193225535967792084153637585725267224878335215248443107599191173632256

Offset: 1

Author

Darse Billings, Oct 11 2009

Keywords

• base
• nonn

Comments

Term a(n) is a cyclical bit string of length 2^n, with every possible substring of length n occurring exactly once.

Mathworld says: "Every de Bruijn sequence corresponds to an Eulerian cycle on a de Bruijn graph. Surprisingly, it turns out that the lexicographic sequence of Lyndon words of lengths divisible by n gives the lexicographically earliest de Bruijn sequence (Ruskey). de Bruijn sequences can be generated by feedback shift registers (Golomb 1967; Ronse 1984; Skiena 1990, p. 196)."

Terms grow like Theta(2^(2^n)). - Darse Billings, Oct 18 2009

Examples

			For n = 3, the last de Bruijn sequence, a(n) = B(2,3), is '11101000' = 232.
		

Links

• Darse Billings, Table of n, a(n) for n=1..9
• Darse Billings, Python program
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
• Eric Weisstein's World of Mathematics, de Bruijn Sequence
• Wikipedia, de Bruijn Sequence

Crossrefs

Cf. A166315 (lexicographically earliest de Bruijn sequences (binary complements)).

Extensions

a(6)-a(8) from Darse Billings, Oct 18 2009