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A171588
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The Pell word: Fixed point of the morphism 0->001, 1->0.
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31
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0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1
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OFFSET
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1,1
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COMMENTS
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This is a Sturmian word: equals the limit word S(infinity) where S(0) = 0, S(1) = 001 and for n >= 1, S(n+1) = S(n)S(n)S(n-1). See the examples below.
This sequence corresponds to the case k = 2 of the Sturmian word S_k(infinity) as defined in A080764. See A159684 for the case k = 1. (End)
Characteristic word with slope 1 - 1/sqrt(2). Since the characteristic word with slope 1-theta is the mirror image of the characteristic word with slope theta, a(n)= 1 - A080764(n) for all n. - Michel Dekking, Jan 31 2017
The positions of 0 comprise A001951 (Beatty sequence for sqrt(2)); the positions of 1 comprise A001952 (Beatty sequence for 2+sqrt(2)). - Clark Kimberling, May 11 2017
This is also the fixed point of the mapping 00->0010, 01->001, 10->010, starting with 00 [Dekking and Keane, 2022]. See A289001. - N. J. A. Sloane, Mar 09 2022
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 284.
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LINKS
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FORMULA
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a(n) = floor((n + 1)/(2 + sqrt(2))) - floor(n /(2 + sqrt(2))). - Peter Bala, Nov 22 2013
a(n) = floor((n+1)(1 - 1/sqrt(2)) - floor(n (1 - 1/sqrt(2)). - Michel Dekking, Jan 31 2017
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EXAMPLE
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The sequence of words S(n) begins
S(0) = 0
S(1) = 001
S(2) = 001 001 0
S(3) = 0010010 0010010 001
S(4) = 00100100010010001 00100100010010001 0010010.
The lengths of the words are [1, 3, 7, 17, 41, ...] = A001333 (apart from the initial term). (End)
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MAPLE
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Digits := 50: u := evalf(2 + sqrt(2)): A171588 := n->floor((n+1)/u) - floor(n/u): seq(A171588(n), n = 1..80); # Peter Bala, Nov 22 2013
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MATHEMATICA
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Table[Floor[(n + 1) (1 - 1/Sqrt[2]) - Floor[n (1 - 1/Sqrt[2])]], {n, 100}] (* Vincenzo Librandi, Jan 31 2017 *)
Nest[Flatten[# /. {0 -> {0, 0, 1}, 1 -> {0}}] &, {0}, 6] (* Clark Kimberling, May 11 2017 *)
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PROG
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(Magma) [Floor((n+1)*(1-1/Sqrt(2))-Floor(n*(1-1/Sqrt(2)))): n in [1..100]]; // Vincenzo Librandi, Jan 31 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Alexis Monnerot-Dumaine (alexis.monnerotdumaine(AT)gmail.com), Dec 12 2009
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STATUS
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approved
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