Search: id:A003849
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%I A003849
%S A003849 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,
%T A003849 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,
%U A003849 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
%N A003849 The infinite Fibonacci word (start with 0, apply 0->01, 1->0, take limit).
%C A003849 A Sturmian word.
%C A003849 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
%C A003849 Characteristic function of A003622 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
May 03 2004
%C A003849 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
%D A003849 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press,
2003.
%D A003849 J. Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull.
EATCS, #79 (2003), pp. 178-228.
%D A003849 S. Ferenczi, Complexity of sequences and dynamical systems, Discrete
Math., 206 (1999), 145-154.
%D A003849 J. Grytczuk, Infinite semi-similar words, Discrete Math. 161 (1996),
133-141.
%D A003849 A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic
Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.
%D A003849 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.
%D A003849 G. Melancon, Factorizing infinite words using Maple, MapleTech journal,
vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
%D A003849 G. Melancon, Lyndon factorization of sturmian words, Discr. Math., 210
(2000), 137-149.
%H A003849 N. J. A. Sloane, Table of n, a(n) for n = 0..10945
%H A003849 Index entries for characteristic functions
%H A003849 J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences.
%H A003849 Jean Berstel, Home Page
%H A003849 C. Kimberling,
A Self-Generating Set and the Golden Mean, J. Integer Sequences,
3 (2000), #00.2.8.
%H A003849 M. Lothaire,
Algebraic Combinatorics on Words, Cambridge, 2002, see p. 41,
etc.
%H A003849 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%F A003849 Define strings S(0)=0, S(1)=01, S(n)=S(n-1)S(n-2); iterate; sequence
is S(infinity).
%F A003849 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
%F A003849 a(n) = A003714(n), mod 2 = A014417(n), mod 2 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Jan 04 2004
%e A003849 The word is 010010100100101001010010010100...
%p A003849 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);
%p A003849 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:
%p A003849 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)
%t A003849 Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (from Robert
G. Wilson v Mar 05 2005)
%o A003849 (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];
%Y A003849 Binary complement of A005614. Cf. A014675, A003842, A036299, A003714,
A014417, A096268, A096270, A133235.
%Y A003849 Positions of 1's gives A003622. A076662 is another version and so are
A003842 and A008352. This one (A003849) is the standard form.
%Y A003849 Sequence in context: A091446 A164349 A094186 this_sequence A115199 A085242
A059620
%Y A003849 Adjacent sequences: A003846 A003847 A003848 this_sequence A003850 A003851
A003852
%K A003849 nonn,easy,nice
%O A003849 0,1
%A A003849 N. J. A. Sloane (njas(AT)research.att.com).
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