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A159684 Sturmian word: limit S(infinity) where S(0) = 0, S(1) = 0,1 and for n>=1, S(n+1) = S(n)S(n)S(n-1). 13
0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Fixed point of morphism 0 -> 0,1; 1 -> 0,1,0.

This sequence corresponds to the case k = 1 of the Sturmian word S_k(infinity) as defined in A080764. See A171588 for the case k = 2. - Peter Bala, Nov 22 2013

This sequence is the {1->01}-transform of the Sturmian word A080764.  - Clark Kimberling, May 17 2017

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.

P. G. Anderson, T. C. Brown, P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity Proc. Amer. Math. Soc. 123 (1995), 2005-2009.

Jeffrey Shallit, Characteristic words as fixed points of homomorphisms, University of Waterloo Technical Report CS-91-72, 1991. See Example 1.

Jeffrey Shallit, Characteristic words as fixed points of homomorphisms. See Example 1. [Cached copy, with permission]

Wikipedia, Sturmian word

FORMULA

From Peter Bala, Nov 22 2013: (Start)

a(n) = floor((n + 2)*(sqrt(2) - 1)) - floor((n + 1)*(sqrt(2) - 1)).

If we read the sequence as the decimal constant C = 0.01010 01010 01010 10010 10010 ... then C = sum {n >= 1} 1/10^floor(n*(1 + sqrt(2))).

The real number 9*C has the simple continued fraction expansion [0; 11, 1010, 10000100, 100000000000100000, 100000000000000000000000000001000000000000, ...], the partial quotients having the form 10^Pell(n)*(1 + 10^Pell(n+1)) = 10^A001333(n+1) + 10^A000129(n) (see Adams and Davison).

A rapidly converging series for C is C = 9*sum {n >= 1} 10^Pell(2*n-1)*(1 + 10^Pell(2*n))/( (10^Pell(2*n-1) - 1)*(10^Pell(2*n+1) - 1) ): for example, the first 10 terms of the series give a rational approximation to C accurate to more than 130 million decimal places. Compare with the Fibonacci words A005614 and A221150. (End)

EXAMPLE

0 -> 0,1 -> 0,1,0,1,0 -> 0,1,0,1,0,0,1,0,1,0,0,1 ->...

MATHEMATICA

Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {0, 1, 0}}] &, {1}, 6] (* Robert G. Wilson v, May 02 2009 *)

PROG

(Haskell)

a159684 n = a159684_list !! n

a159684_list = 0 : concat (iterate (concatMap s) [1])

   where s 0 = [0, 1]; s 1 = [0, 1, 0]

-- Reinhard Zumkeller, Oct 26 2013

CROSSREFS

See A188037 for another version of this sequence. - N. J. A. Sloane, Mar 22 2011

Cf. A000129, A001333, A005614, A080764, A119812, A171588, A221150, A221151, A221152, A096270.

Sequence in context: A092202 A285686 A303591 * A244221 A253050 A241575

Adjacent sequences:  A159681 A159682 A159683 * A159685 A159686 A159687

KEYWORD

nonn,easy

AUTHOR

Philippe Deléham, Apr 19 2009

EXTENSIONS

More terms from Robert G. Wilson v, May 02 2009

STATUS

approved

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Last modified October 18 14:52 EDT 2019. Contains 328161 sequences. (Running on oeis4.)