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A016031 De Bruijn's sequence: 2^(2^(n-1) - n): ways of arranging 2^n bits in circle so all 2^n consecutive strings of length n are distinct. 14
1, 1, 2, 16, 2048, 67108864, 144115188075855872, 1329227995784915872903807060280344576, 226156424291633194186662080095093570025917938800079226639565593765455331328 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Sequence corresponds also to the largest number that may be determined by asking no more than 2^(n-1) - 1 questions("Yes"-or-"No" answerable) with lying allowed at most once. - Lekraj Beedassy, Jul 15 2002

Number of Ouroborean rings for binary n-tuplets. - Lekraj Beedassy, May 06 2004

Also the number of games of Nim that are wins for the second player when the starting position is either the empty heap or heaps of sizes 1 <= t_1 < t_2 < ... < t_k < 2^(n-1) (if n is 1, the only starting position is the empty heap). E.g.: a(4) = 16: the winning positions for the second player when all the heap sizes are different and less than 2^3: (4,5,6,7), (3,5,6), (3,4,7), (2,5,7), (2,4,6), (2,3,6,7), (2,3,4,5), (1,6,7), (1,4,5), (1,3,5,7), (1,3,4,6), (1,2,5,6), (1,2,4,7), (1,2,3), (1,2,3,4,5,6,7) and the empty heap. - Kennan Shelton (kennan.shelton(AT)gmail.com), Apr 14 2006

a(n + 1) = 2^(2^n-n-1) = 2^A000295(n) is also the number of set-systems on n vertices with no singletons. The case with singletons is A058891. The unlabeled case is A317794. The spanning/covering case is A323816. The antichain case is A006126. The connected case is A323817. The uniform case is A306021(n) - 1. The graphical case is A006125. The chain case is A005840. - Gus Wiseman, Feb 01 2019

REFERENCES

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 255.

F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 31.

D. J. Newman, "A variation of the Game of Twenty Question", Prob. 66-20 pp. 121-2 In Problems in Applied Mathematics, Ed. M. S. Klamkin, SIAM PA 1990.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Cor. 5.6.15.

I. Stewart, Game, Set and Math, pp. 50, Penguin 1991.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..12

CombOS - Combinatorial Object Server, Generate de Bruijn sequences

R. Erra, N. Lygeros and N. Stewart, On Minimal Strings Containing the Elements of S(n) by Decimation, Proceedings AA (DM-CCG), 2001, Section 5.4.

Wikipedia, De Bruijn sequence

FORMULA

a(n) = 2^A000295(n-1). - Lekraj Beedassy, Jan 17 2007

Shifting once to the left gives the binomial transform of A323816. - Gus Wiseman, Feb 01 2019

MAPLE

P:=proc(n) local i, j; for i from 1 by 1 to n do j:=2^(2^(i-1)-i); print(j); od; end: P(20); # Paolo P. Lava, May 11 2006

MATHEMATICA

Table[2^(2^(n - 1) - n), {n, 20}] (* Vincenzo Librandi, Aug 09 2017 *)

PROG

(MAGMA) [2^(2^(n-1)-n): n in [1..10]]; // Vincenzo Librandi, Aug 09 2017

CROSSREFS

Cf. A000295, A003465, A006125, A058891 (set systems), A317794 (unlabeled case), A323816 (spanning case), A323817 (connected case).

Sequence in context: A326974 A060597 A091479 * A001309 A132569 A165644

Adjacent sequences:  A016028 A016029 A016030 * A016032 A016033 A016034

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

STATUS

approved

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Last modified October 14 11:36 EDT 2019. Contains 327996 sequences. (Running on oeis4.)