Search: id:A010059
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%I A010059
%S A010059 1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0,1,1,
%T A010059 0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,
%U A010059 0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0
%N A010059 Another version of the Thue-Morse sequence: let A_k denote the first 
               2^k terms; then A_0 = 1 and for k >= 0, A_{k+1} = A_k B_k, where 
               B_k is obtained from A_k by interchanging 0's and 1's.
%C A010059 Characteristic function of A001969 (evil numbers). - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Jun 20 2003
%C A010059 a(n)+A010060(n)=1 for all n.
%C A010059 a(n) = A159481(n+1) - A159481(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 16 2009]
%D A010059 Dejean, F.; Sur un theoreme de Thue. J. Combinatorial Theory Ser. A 13 
               (1972), 90-99.
%D A010059 W. H. Gottschalk and G. A. Hedlund, Topological Dynamics. American Mathematical 
               Society, Colloquium Publications, Vol. 36, Providence, RI, 1955, 
               p. 105.
%D A010059 G. A. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk 
               Mat. Tid., 15 (1967), 148-150.
%D A010059 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, 
               p. 23.
%D A010059 M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. 
               Amer. Math. Soc., 22 (1921), 84-100.
%D A010059 A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, 
               Rockville, MD, 1981, p. 6.
%H A010059 <a href="Sindx_Ch.html#char_fns">Index entries for characteristic functions</
               a>
%H A010059 J.-P. Allouche and J. O. Shallit, <a href="http://www.cs.uwaterloo.ca/
               ~shallit/Papers/ubiq.ps">The Ubiquitous Prouhet-Thue-Morse Sequence</
               a>, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences 
               and Their Applications: Proceedings of SETA '98, Springer-Verlag, 
               1999, pp. 1-16.
%H A010059 Michael Gilleland, <a href="selfsimilar.html">Some Self-Similar Integer 
               Sequences</a>
%H A010059 M. Morse, <a href="http://links.jstor.org/sici?sici=0002-9947%28192101%2922%3A1%3C84%3ARGOASO%3E2.0.CO%3B2-9"\
               >Recurrent geodesics on a surface of negative curvature</a> (page 
               images), Trans. Amer. Math. Soc., 22 (1921), 84-100.
%H A010059 Stephen Wolfram, <a href="http://www.wolframscience.com/nksonline/page-889c-text?firstview=1">
               A New Kind Of Science | Online</a>.
%F A010059 G.f.: 1/2 * (1/(1-x) + prod(k>=0, 1-x^2^k)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Jun 20 2003
%e A010059 The evolution starting at 1 is:
%e A010059 .1
%e A010059 .1, 0
%e A010059 .1, 0, 0, 1,
%e A010059 .1, 0, 0, 1, 0, 1, 1, 0
%e A010059 .1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1
%e A010059 .1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 
               1, 1, 0, 0, 1, 0, 1, 1, 0
%e A010059 ...........
%p A010059 A010059 := n->1-A010060(n);
%t A010059 Mod[ CoefficientList[Series[(1 + Sqrt[(1 - 3x)/(1 + x)])/(2(1 + x)), 
               {x, 0, 111}], x], 2] (from Stephan Wolfram)
%t A010059 CoefficientList[ Series[1/(1 - x) + Product[1 - x^2^k, {k, 0, 10}], {x, 
               0, 111}]/2, x] (from Robert G. Wilson v Jul 16 2004)
%t A010059 Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {1}, 7] (* Robert 
               G. Wilson v Sep 26 2006)
%Y A010059 Cf. A001285 (1, 2 version), A010060 (0, 1 version).
%Y A010059 Sequence in context: A114591 A005171 A076404 this_sequence A143580 A011749 
               A104105
%Y A010059 Adjacent sequences: A010056 A010057 A010058 this_sequence A010060 A010061 
               A010062
%K A010059 nonn
%O A010059 0,1
%A A010059 N. J. A. Sloane (njas(AT)research.att.com).

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