Search: id:A005614
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%I A005614
%S A005614 1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,
%T A005614 0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,
%U A005614 1,0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0
%N A005614 Infinite Fibonacci word (start with 1, apply 0->1, 1->10, iterate).
%C A005614 Characteristic function of A022342 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               May 03 2004
%C A005614 a(n)=number of 0's between successive 1's (see also A003589 and A007538) 
               - Eric Angelini (eric.angelini(AT)kntv.be), Jul 06 2005
%C A005614 With offset 1 this is the characteristic sequence for Wythoff A-numbers 
               A000201=[1,3,4,6,...].
%D A005614 F. Axel et al., Vibrational modes in a one dimensional "quasi-alloy": 
               the Morse case, J. de Physique, Colloq. C3, Supp. to No. 7, Vol. 
               47 (Jul 1986), pp. C3-181-C3-186; see Eq. (10).
%D A005614 M. Bunder and K. Tognetti, On the self matching properties of [j tau], 
               Discrete Math., 241 (2001), 139-151.
%D A005614 S. Dulucq and D. Gouyou-Beauchamps, Sur les facteurs des suites de Sturm, 
               Theoret. Comput. Sci. 71 (1990), 381-400.
%D A005614 M. S. El Naschie, Statistical geometry of a Cantor discretum and semiconductors, 
               Computers Math. Applic., 29 (No, 12, 1995), 103-110.
%D A005614 J. Grytczuk, Infinite semi-similar words, Discrete Math. 161 (1996), 
               133-141.
%D A005614 G. Melancon, Factorizing infinite words using Maple, MapleTech journal, 
               vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
%D A005614 G. Melancon, Lyndon factorization of sturmian words, Discr. Math., 210 
               (2000), 137-149.
%H A005614 T. D. Noe, <a href="b005614.txt">Table of n, a(n) for n=0..10945</a> 
               (19 iterations)
%H A005614 <a href="Sindx_Ch.html#char_fns">Index entries for characteristic functions</
               a>
%H A005614 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A005614 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
               fibrab.html">The Fibonacci Rabbit Sequence</a>
%H A005614 J. O. Shallit, <a href="http://www.cs.uwaterloo.ca/cs-archive/CS-1991/
               72/fix0.pdf">Characteristic words as fixed points of homomorphisms</
               a>
%H A005614 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RabbitConstant.html">Link to a section of The World of Mathematics.</
               a>
%H A005614 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RabbitSequence.html">Link to a section of The World of Mathematics.</
               a>
%F A005614 Define strings S(0)=1, S(1)=10, S(n)=S(n-1)S(n-2); iterate.
%F A005614 floor((n+1)*u)-floor(n*u), u = (1-sqrt(5))/2.
%e A005614 The infinite word is 101101011011010110101101101011...
%p A005614 Digits := 50; u := evalf((1-sqrt(5))/2); A005614 := n->floor((n+1)*u)-floor(n*u);
%t A005614 Nest[ Function[l, {Flatten[(l /. {0 -> {1}, 1 -> {1, 0}})]}], {1}, 10] 
               (from Robert G. Wilson v Jan 30, 2005)
%o A005614 (PARI) a(n,w1,s0,s1)=local(w2); for(i=2,n,w2=[ ]; for(k=1,length(w1),
               w2=concat(w2, if(w1[ k ],s1,s0))); w1=w2); w2
%o A005614 (PARI) for(n=2,10,print(n" "a(n,[ 0 ],[ 1 ],[ 1,0 ]))) \\ Gives successive 
               convergents to sequence
%Y A005614 Binary complement of A003849, which is the standard form of this sequence. 
               Cf. A036299, A001468, A014675.
%Y A005614 Two other essentially identical sequences are A096270, A114986.
%Y A005614 Sequence in context: A096055 A125144 A115198 this_sequence A071036 A166946 
               A141687
%Y A005614 Adjacent sequences: A005611 A005612 A005613 this_sequence A005615 A005616 
               A005617
%K A005614 nonn,easy,nice
%O A005614 0,1
%A A005614 N. J. A. Sloane (njas(AT)research.att.com).
%E A005614 Corrected by Clark Kimberling (ck6(AT)evansville.edu), Oct 04 2000

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