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A001285
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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.
(Formerly M0193 N0071)
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49
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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, 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, 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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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Or, follow a(0), .., a(2^k-1) by its complement.
Equals limiting row of A161175. [From Gary W. Adamson, Jun 05 2009]
Parse A010060 into consecutive pairs: (01, 10, 10, 01, 10, 01,...); then apply the rules: (01 -> 1; 10 ->2), obtaining (1, 2, 2, 1, 2, 1, 1,...). - Gary W. Adamson, Oct 25 2010
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 15.
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).
F. Dejean, Sur un theoreme de Thue. J. Combinatorial Theory Ser. A 13 (1972), 90-99.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
W. H. Gottschalk and G. A. Hedlund, Topological Dynamics. American Mathematical Society, Colloquium Publications, Vol. 36, Providence, RI, 1955, p. 105.
G. A. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk Mat. Tid., 15 (1967), 148-150.
A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 23.
M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100.
A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 6.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1023
J.-P. Allouche and __Jeffrey Shallit__, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
Michael Gilleland, Some Self-Similar Integer Sequences
N. J. A. Sloane, The first 1000 terms as a string
S. Wolfram, Source for short Thue-Morse generating code
Index entries for "core" sequences
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FORMULA
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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.
a(n) = 2-A010059(n) = 1/2*(3-(-1)^A000120(n)). - Ralf Stephan, Jun 20 2003
a(n)=sum(k=0, n, binomial(n, k){mod 2}) {mod 3}=A001316(n) {mod 3} - Benoit Cloitre, May 09 2004
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MAPLE
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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;
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
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MATHEMATICA
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Nest[ Flatten@ Join[#, # /. {1 -> 2, 2 -> 1}] &, {1}, 7] (* from Robert G. Wilson v Feb 26 2005 *)
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PROG
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(PARI) a(n)=1+subst(Pol(binary(n)), x, 1)%2
(PARI) a(n)=sum(k=0, n, binomial(n, k)%2)%3
(PARI) a(n)=hammingweight(n)%2+1 \\ Charles R Greathouse IV, Mar 26 2013
(Haskell)
a001285 n = a001285_list !! n
a001285_list = map (+ 1) a010060_list
-- Reinhard Zumkeller, Oct 03 2012
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CROSSREFS
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Cf. A010060 for 0, 1 version, which is really the main entry for this sequence; also A003159. A001285(n)=1+A010060(n), A225186 (squares).
A026465 gives run lengths.
Cf. A010059 (1, 0 version).
Cf. A161175 [From Gary W. Adamson, Jun 05 2009]
Sequence in context: A071292 A088569 A192763 * A088424 A097456 A164002
Adjacent sequences: A001282 A001283 A001284 * A001286 A001287 A001288
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KEYWORD
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nonn,easy,core,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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