0,1

A Sturmian word.

Replace each run (1;1) with (1;0) in 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 (benoit7848c(AT)orange.fr), Nov 10 2003

Characteristic function of A003622 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 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 (njas(AT)research.att.com), Sep 24 2007

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

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

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.

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.

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

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

Index entries for characteristic functions

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

Jean Berstel, Home Page

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.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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

a(n) = floor((n+2)*r)-floor((n+1)*r) where r=phi/(1+2*phi) and phi is the Golden Ratio. - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2003

a(n) = A003714(n), mod 2 = A014417(n), mod 2 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 04 2004

The word is 010010100100101001010010010100...

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)

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

(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];

Binary complement of A005614. Cf. A014675, A003842, A036299, A003714, A014417, A096268, A096270, A133235.

Positions of 1's gives A003622. A076662 is another version and so are A003842 and A008352. This one (A003849) is the standard form.

Adjacent sequences: A003846 A003847 A003848 this_sequence A003850 A003851 A003852

Sequence in context: A091446 A164349 A094186 this_sequence A115199 A085242 A059620

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).