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Search: id:A010059
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| A010059 |
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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. |
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+0
23
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| 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, 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, 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
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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Characteristic function of A001969 (evil numbers). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 20 2003
a(n)+A010060(n)=1 for all n.
a(n) = A159481(n+1) - A159481(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 16 2009]
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REFERENCES
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Dejean, F.; Sur un theoreme de Thue. J. Combinatorial Theory Ser. A 13 (1972), 90-99.
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.
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.
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LINKS
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Index entries for characteristic functions
J.-P. Allouche and J. O. 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
M. Morse, Recurrent geodesics on a surface of negative curvature (page images), Trans. Amer. Math. Soc., 22 (1921), 84-100.
Stephen Wolfram, A New Kind Of Science | Online.
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FORMULA
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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
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EXAMPLE
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The evolution starting at 1 is:
.1
.1, 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
.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
...........
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MAPLE
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A010059 := n->1-A010060(n);
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MATHEMATICA
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Mod[ CoefficientList[Series[(1 + Sqrt[(1 - 3x)/(1 + x)])/(2(1 + x)), {x, 0, 111}], x], 2] (from Stephan Wolfram)
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)
Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {1}, 7] (* Robert G. Wilson v Sep 26 2006)
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CROSSREFS
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Cf. A001285 (1, 2 version), A010060 (0, 1 version).
Sequence in context: A114591 A005171 A076404 this_sequence A143580 A011749 A104105
Adjacent sequences: A010056 A010057 A010058 this_sequence A010060 A010061 A010062
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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