

A166315


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


3




OFFSET

1,2


COMMENTS

Term a(n) is a cyclical bit string of length 2^n, with every possible substring of length n occurring exactly once.
Mathworld (http://mathworld.wolfram.com/deBruijnSequence.html) 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 smallest 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


LINKS

Darse Billings, Table of n, a(n) for n=1..9
Darse Billings, Python program
Mathworld, de Bruijn Sequence
F. Ruskey, Information on necklaces, unlabelled necklaces, Lyndon words, de Bruijn sequences
Wikipedia, de Bruijn Sequence


EXAMPLE

Example: For n = 3, the first de Bruijn sequence, a(n) = B(2,3), is '00010111' = 23.


CROSSREFS

Cf. A166316 = 2, 12, 232, 63056, 4221224224, ..., Lexicographically largest de Bruijn sequences (binary complements).
Sequence in context: A009593 A009818 A210734 * A113577 A224700 A204578
Adjacent sequences: A166312 A166313 A166314 * A166316 A166317 A166318


KEYWORD

base,nonn


AUTHOR

Darse Billings, Oct 11 2009


EXTENSIONS

Added three more terms, a(6)a(8) Darse Billings, Oct 18 2009


STATUS

approved



