Search: id:A001285
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%I A001285 M0193 N0071
%S A001285 1,2,2,1,2,1,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,2,2,1,1,2,2,1,2,1,1,2,2,1,
%T A001285 1,2,1,2,2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,1,1,2,2,1,1,2,1,2,2,1,2,1,1,2,
%U A001285 1,2,2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,1,1,2,2,1,1,2,1,2,2,1,1,2,2,1,2,1
%N A001285 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 1's and 2's.
%C A001285 Or, follow a(0), .., a(2^k-1) by its complement.
%C A001285 Equals convergent as an infinite string of A161175 row terms. [From Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Jun 05 2009]
%D A001285 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 
               2003, p. 15.
%D A001285 F. Axel et al., Vibrational modes in a one dimensional "quasi-alloy": 
               the Morse case, J. de Physique, Colloq. C3, Supp. to No. 7, Vol. 
               47 (Jul 1986), pp. C3-181-C3-186; see Eq. (10).
%D A001285 F. Dejean, Sur un theoreme de Thue. J. Combinatorial Theory Ser. A 13 
               (1972), 90-99.
%D A001285 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence 
               Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A001285 W. H. Gottschalk and G. A. Hedlund, Topological Dynamics. American Mathematical 
               Society, Colloquium Publications, Vol. 36, Providence, RI, 1955, 
               p. 105.
%D A001285 G. A. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk 
               Mat. Tid., 15 (1967), 148-150.
%D A001285 A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic 
               Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.
%D A001285 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, 
               p. 23.
%D A001285 M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. 
               Amer. Math. Soc., 22 (1921), 84-100.
%D A001285 A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, 
               Rockville, MD, 1981, p. 6.
%D A001285 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001285 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001285 T. D. Noe, <a href="b001285.txt">Table of n, a(n) for n = 0..1023</a>
%H A001285 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 A001285 Michael Gilleland, <a href="selfsimilar.html">Some Self-Similar Integer 
               Sequences</a>
%H A001285 S. Wolfram, <a href="http://www.wolframscience.com/nksonline/page-889c-text">
               Source for short Thue-Morse generating code</a>
%H A001285 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001285 a(2n)=a(n), a(2n+1)=3-a(n), a(0)=1. Also, a(k+2^m)=3-a(k) if 0<=k<2^m.
%F A001285 a(n) = 2-A010059(n) = 1/2*(3-(-1)^A000120(n)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Jun 20 2003
%F A001285 a(n)=sum(k=0, n, binomial(n, k){mod 2}) {mod 3}=A001316(n) {mod 3} - 
               Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2004
%p A001285 A001285 := proc(n) option remember; if n=0 then 1 elif n mod 2 = 0 then 
               A001285(n/2) else 3-A001285((n-1)/2); fi; end;
%p A001285 s := proc(k) local i, ans; ans := [ 1,2 ]; for i from 0 to k do ans := 
               [ op(ans),op(map(n->if n=1 then 2 else 1 fi, ans)) ] od; RETURN(ans); 
               end; t1 := s(6); A001285 := n->t1[n]; # s(k) gives first 2^(k+2) 
               terms
%t A001285 Nest[ Function[l, {Flatten[(l /. {2 -> {2, 1}, 1 -> {1, 2}})]}], {1}, 
               7] (from Robert G. Wilson v Feb 26 2005)
%o A001285 (PARI) a(n)=1+subst(Pol(binary(n)),x,1)%2
%o A001285 (PARI) a(n)=sum(k=0,n,binomial(n,k)%2)%3
%Y A001285 Cf. A010060 (for 0, 1 version), A003159. A001285(n)=1+A010060(n).
%Y A001285 A026465 gives run lengths.
%Y A001285 Cf. A010059 (1, 0 version).
%Y A001285 A161175 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 05 2009]
%Y A001285 Sequence in context: A035214 A071292 A088569 this_sequence A088424 A097456 
               A164002
%Y A001285 Adjacent sequences: A001282 A001283 A001284 this_sequence A001286 A001287 
               A001288
%K A001285 nonn,easy,core,nice
%O A001285 0,2
%A A001285 N. J. A. Sloane (njas(AT)research.att.com).

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