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A080764 First differences of A049472, floor(n/sqrt(2)). 9
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Fixed point of the morphism 0->{1}, 1->{1,1,0}. - Benoit Cloitre, May 31 2004

As binary constant 0.1101101110110... = 0.85826765646... (A119812), see Fxtbook link. - Joerg Arndt, May 15 2011

Characteristic word with slope 1/sqrt(2) [J. L. Ramirez et al., arXiv:1212.1368]. - R. J. Mathar, Jul 09 2013

From Peter Bala, Nov 22 2013: (Start)

Sturmian word: equals the limit word S(infinity) where S(0) = 0, S(1) = 1 and for n >= 1, S(n+1) = S(n)S(n)S(n-1).

More generally, for k = 0,1,2,..., we can define a sequence of words S_k(n) by S_k(0) = 0, S_k(1) = 0...01 (k 0's) and for n >= 1, S_k(n+1) = S_k(n)S_k(n)S_k(n-1). Then the limit word S_k(infinity) is a Sturmian word whose terms are given by a(n) = floor((n + 2)/(k + sqrt(2))) - floor((n + 1)/(k + sqrt(2))).

This sequence corresponds to the case k = 0. See A159684 (case k = 1) and A171588 (case k = 2). Compare with the Fibonacci words A005614, A221150, A221151 and A221152. See also A230901. (End)

LINKS

Table of n, a(n) for n=0..104.

Joerg Arndt, Matters Computational (The Fxtbook), section 38.12, pp. 757-758

Wikipedia, Sturmian word

EXAMPLE

From Peter Bala, Nov 22 2013: (Start)

The first few Sturmian words S(n) are

S(0) = 0

S(1) = 1

S(2) = 110

S(3) = 110 110 1

S(4) = 1101101 1101101 110

S(5) = 11011011101101110 11011011101101110 1101101

The lengths of the words are [1, 1, 3, 7, 17, 41, ...] = A001333.  (End)

MAPLE

A080764 := proc(n)

    alpha := 1/sqrt(2) ;

    floor((n+2)*alpha)-floor((n+1)*alpha) ;

end proc: # R. J. Mathar, Jul 09 2013

MATHEMATICA

Nest[ Flatten[ # /. {0 -> 1, 1 -> {1, 1, 0}}] &, {1}, 7] (* Robert G. Wilson v, Apr 16 2005 *)

NestList[ Flatten[ # /. {0 -> {1}, 1 -> {1, 0, 1}}] &, {1}, 5] // Flatten (* or *)

t = Table[Floor[n/Sqrt[2]], {n, 111}]; Drop[t, 1] - Drop[t, -1] (* Robert G. Wilson v, Nov 03 2005 *)

CROSSREFS

A005614, A159684, A171588, A221150, A221151, A221152, A230901.

Sequence in context: A015923 A014768 A015527 * A014114 A014219 A065828

Adjacent sequences:  A080761 A080762 A080763 * A080765 A080766 A080767

KEYWORD

nonn,easy,changed

AUTHOR

Matthew Vandermast, Mar 25 2003

STATUS

approved

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Last modified November 28 11:48 EST 2014. Contains 250343 sequences.