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A016031 De Bruijn's sequence: 2^(2^(n-1) - n): number of ways of arranging 2^n bits in circle so all 2^n consecutive strings of length n are distinct. 37


%S 1,1,2,16,2048,67108864,144115188075855872,

%T 1329227995784915872903807060280344576,

%U 226156424291633194186662080095093570025917938800079226639565593765455331328

%N De Bruijn's sequence: 2^(2^(n-1) - n): number of ways of arranging 2^n bits in circle so all 2^n consecutive strings of length n are distinct.

%C 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

%C Number of Ouroborean rings for binary n-tuplets. - _Lekraj Beedassy_, May 06 2004

%C 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

%C 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

%C Named after the Dutch mathematician Nicolaas Govert de Bruijn (1918-2012). - _Amiram Eldar_, Jun 20 2021

%D Jonathan L. Gross and Jay Yellen, eds., Handbook of Graph Theory, CRC Press, 2004, p. 255.

%D Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973, p. 31.

%D D. J. Newman, "A variation of the Game of Twenty Question", in: Murray S. Klamkin (ed.), Problems in Applied Mathematics, Philadelphia, PA: SIAM, 1990, Problem 66-20, pp. 121-122.

%D Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Cor. 5.6.15.

%D Ian Stewart, Game, Set and Math, pp. 50, Penguin 1991.

%H Vincenzo Librandi, <a href="/A016031/b016031.txt">Table of n, a(n) for n = 1..12</a>

%H CombOS - Combinatorial Object Server, <a href="http://combos.org/bruijn">Generate de Bruijn sequences</a>.

%H Robert Erra, Nik Lygeros and Nigel Stewart, <a href="http://www.dmtcs.org/pdfpapers/dmAA0112.pdf">On Minimal Strings Containing the Elements of S(n) by Decimation</a>, Proceedings AA (DM-CCG), 2001, Section 5.4.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/De_Bruijn_sequence">De Bruijn sequence</a>.

%F a(n) = 2^A000295(n-1). - _Lekraj Beedassy_, Jan 17 2007

%F Shifting once to the left gives the binomial transform of A323816. - _Gus Wiseman_, Feb 01 2019

%p 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

%t Table[2^(2^(n - 1) - n), {n, 20}] (* _Vincenzo Librandi_, Aug 09 2017 *)

%o (MAGMA) [2^(2^(n-1)-n): n in [1..10]]; // _Vincenzo Librandi_, Aug 09 2017

%Y Cf. A000295, A003465, A006125, A058891 (set systems), A317794 (unlabeled case), A323816 (spanning case), A323817 (connected case), A331691 (alternating signs).

%K nonn,easy,nice

%O 1,3

%A _N. J. A. Sloane_, _Robert G. Wilson v_

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Last modified July 3 17:54 EDT 2022. Contains 355055 sequences. (Running on oeis4.)