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A003849 The infinite Fibonacci word (start with 0, apply 0->01, 1->0, take limit). 72
0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

A Sturmian word.

Define strings S(0)=0, S(1)=01, S(n)=S(n-1)S(n-2); iterate; sequence is S(infinity).

The 0's occur at positions in A022342 (i.e., A000201 - 1), the 1's at positions in A003622.

Replace each run (1;1) with (1;0) in the infinite Fibonacci word A005614 (and add 0 as prefix) A005614 begins : 1,0,1,1,0,1,0,1,1,0,1,1,... changing runs (1,1) with (1,0) produces 1,0,0,1,0,1,0,0,1,0,0,1,... - Benoit Cloitre, Nov 10 2003

Characteristic function of A003622. - Philippe Deléham, May 03 2004

The fraction of 0's in the first n terms approaches 1/phi (see for example Allouche and Shallit). - N. J. A. Sloane, Sep 24 2007

The limiting mean and variance of the first n terms are both equal to 2-phi. - Clark Kimberling, Mar 12 2014

REFERENCES

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.

J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc. - see p. 64.

G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..10945

A. G. M. Ahmed, AA Weaving, in Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture.

J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences.

P. Arnoux and E. Harriss, What is a Rauzy Fractal?, Notices Amer. Math. Soc., 61 (No. 7, 2014), 768-770, also p. 704 and front cover.

Jean Berstel, Home Page

J. Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull. EATCS, #79 (2003), pp. 178-228.

Fabien Durand, Julien Leroy, and Gwenaël Richomme, Do the Properties of an S-adic Representation Determine Factor Complexity?, Journal of Integer Sequences, Vol. 16 (2013), #13.2.6.

J. Endrullis, D. Hendriks and J. W. Klop, Degrees of streams.

S. Ferenczi, Complexity of sequences and dynamical systems, Discrete Math., 206 (1999), 145-154.

J. Grytczuk, Infinite semi-similar words, Discrete Math. 161 (1996), 133-141.

A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.

C. Kimberling, A Self-Generating Set and the Golden Mean, J. Integer Sequences, 3 (2000), #00.2.8.

M. Lothaire, Algebraic Combinatorics on Words, Cambridge, 2002, see p. 41, etc.

G. Melancon, Lyndon factorization of sturmian words, Discr. Math., 210 (2000), 137-149.

F. Mignosi, A. Restivo, M. Sciortino, Words and forbidden factors, WORDS (Rouen, 1999). Theoret. Comput. Sci. 273 (2002), no. 1-2, 99--117. MR1872445 (2002m:68096) - From N. J. A. Sloane, Jul 10 2012

Kerry Mitchell, Spirolateral-Type Images from Integer Sequences, 2013

Kerry Mitchell, Spirolateral image for this sequence [taken, with permission, from the Spirolateral-Type Images from Integer Sequences article]

T. D. Noe, The first 1652 subwords, including leading zeros.

Giuseppe Pirillo, Fibonacci numbers and words, Discrete Math. 173 (1997), no. 1-3, 197--207. MR1468849 (98g:68135)

J. L. Ramírez, G. N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014).

José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368, 2012.

M. Rigo, P. Salimov, and E. Vandomme, Some Properties of Abelian Return Words, Journal of Integer Sequences, Vol. 16 (2013), #13.2.5.

Eric Weisstein's World of Mathematics, Golden Ratio

Index entries for characteristic functions

FORMULA

a(n) = floor((n+2)*r) - floor((n+1)*r) where r=phi/(1+2*phi) and phi is the Golden Ratio. - Benoit Cloitre, Nov 10 2003

a(n) = A003714(n) mod 2 = A014417(n) mod 2. - Philippe Deléham, Jan 04 2004

EXAMPLE

The word is 010010100100101001010010010100...

Over the alphabet {a,b} this is a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, ...

MAPLE

z := proc(m) option remember; if m=0 then [0] elif m=1 then [0, 1] else [op(z(m-1)), op(z(m-2))]; fi; end; z(12);

M:=19; S[0]:=`0`; S[1]:=`01`; for n from 2 to M do S[n]:=cat(S[n-1], S[n-2]); od:

t0:=S[M]: l:=length(t0); for i from 1 to l do lprint(i-1, substring(t0, i..i)); od: # N. J. A. Sloane, Nov 01 2006

MATHEMATICA

Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (* Robert G. Wilson v, Mar 05 2005 *)

Flatten[Nest[{#, #[[1]]} &, {0, 1}, 9]] (* IWABUCHI Yu(u)ki, Oct 23 2013 *)

PROG

(MAGMA) t1:=[ n le 2 select ["0", "0, 1"][n] else Self(n-1) cat ", " cat Self(n-2) : n in [1..12]]; t1[12];

(Haskell)

a003849 n = a003849_list !! n

a003849_list = tail $ concat fws where

   fws = [1] : [0] : (zipWith (++) fws $ tail fws)

-- Reinhard Zumkeller, Nov 01 2013, Apr 07 2012

(PARI) a(n)=my(k=2); while(fibonacci(k)<=n, k++); while(n>1, while(fibonacci(k--)>n, ); n-=fibonacci(k)); n==1 \\ Charles R Greathouse IV, Feb 03 2014

CROSSREFS

There are several versions of this sequence in the OEIS. This one and A003842 are probably the most important. See also A008352, A076662.

Binary complement of A005614. Cf. A014675, A036299, A003714, A014417, A096268, A096270, A133235, A182028, A213975.

Positions of 1's gives A003622.

Sequence in context: A091446 A164349 A094186 * A115199 A085242 A059620

Adjacent sequences:  A003846 A003847 A003848 * A003850 A003851 A003852

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Revised by N. J. A. Sloane, Jul 03 2012

STATUS

approved

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Last modified September 1 08:08 EDT 2015. Contains 261265 sequences.