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 A005614 The binary complement of the infinite Fibonacci word A003849. Start with 1, apply 0->1, 1->10, iterate, take limit. 80

%I

%S 1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,0,

%T 1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,0,

%U 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 The binary complement of the infinite Fibonacci word A003849. Start with 1, apply 0->1, 1->10, iterate, take limit.

%C Previous name was: The infinite Fibonacci word (start with 1, apply 0->1, 1->10, iterate, take limit).

%C Characteristic function of A022342. - _Philippe Deléham_, May 03 2004

%C a(n) = number of 0's between successive 1's (see also A003589 and A007538). - _Eric Angelini_, Jul 06 2005

%C With offset 1 this is the characteristic sequence for Wythoff A-numbers A000201=[1,3,4,6,...].

%C Eric Angelini's comment made me think that if 1 is defined as the number of 0's between successive 1's in a string of 0's and 1's, then this string is 101. Applying the same operation to the digits of 101 leads to 101101, the iteration leads to successive palindromes of lengths given by A001911, up to a(n). - _Rémi Schulz_, Jul 06 2010

%C For generalized Fibonacci words see A221150, A221151, A221152... - _Peter Bala_, Nov 11 2013

%C The limiting mean of the first n terms is phi - 1; the limiting variance is phi (A001622). - _Clark Kimberling_, Mar 12 2014

%C Apply the difference operator to every column of the Wythoff difference array, A080164, to get an array of Fibonacci numbers, F(h). Replace each F(h) with h, and apply the difference operator to every column. In the resulting array, every column is A005614. - _Clark Kimberling_, Mar 02 2015

%C Binary expansion of the rabbit constant A014565. - _M. F. Hasler_, Nov 10 2018

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

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

%H T. D. Noe, <a href="/A005614/b005614.txt">Table of n, a(n) for n = 0..10945</a> (19 iterations)

%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2009.02669">Symbolic dynamical scales: modes, orbitals, and transversals</a>, arXiv:2009.02669 [math.DS], 2020.

%H F. Axel et al., <a href="http://dx.doi.org/10.1051/jphyscol:1986318">Vibrational modes in a one dimensional "quasi-alloy": the Morse case</a>, J. de Physique, Colloq. C3, Supp. to No. 7, Vol. 47 (Jul 1986), pp. C3-181-C3-186; see Eq. (10).

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, 753-754.

%H E. A. Bender and J. T. Butler, <a href="/A005612/a005612.pdf">Asymptotic approximations for the number of fanout-free functions</a>, IEEE Trans. Computers, 27 (1978), 1180-1183. (Annotated scanned copy)

%H M. Bunder and K. Tognetti, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00147-9">On the self matching properties of [j tau]</a>, Discrete Math., 241 (2001), 139-151.

%H J. T. Butler, <a href="/A005607/a005607_1.pdf">Letter to N. J. A. Sloane, Dec. 1978</a>.

%H Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, Daisy Ann A. Disu, <a href="http://www.dmmmsu-sluc.com/wp-content/uploads/2018/03/CAS-Monitor-2016-2017-1.pdf">On Fractal Sequences</a>, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.

%H S. Dulucq and D. Gouyou-Beauchamps, <a href="http://dx.doi.org/10.1016/0304-3975(90)90050-R">Sur les facteurs des suites de Sturm</a>, Theoret. Comput. Sci. 71 (1990), 381-400.

%H M. S. El Naschie, <a href="http://dx.doi.org/10.1016/0898-1221(95)00062-4">Statistical geometry of a Cantor discretum and semiconductors</a>, Computers & Mathematics with Applications, Vol. 29 (Issue 12, June 1995), 103-110.

%H D. Gault and M. Clint, <a href="http://dx.doi.org/10.1080/00207168808803682">"Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function</a>, Internat. J. Computer Math., 26 (1988), 35-43.

%H D. Gault & M. Clint, <a href="/A005206/a005206.pdf">"Curiouser and curiouser said Alice. Further reflections on an interesting recursive function</a>, Intern. J. Computer. Math., 26 (1988), 35-43. (Annotated scanned copy)

%H J. Grytczuk, <a href="http://dx.doi.org/10.1016/0012-365X(95)00297-A">Infinite semi-similar words</a>, Discrete Math. 161 (1996), 133-141.

%H C. Kimberling and K. B. Stolarsky, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.3.267">Slow Beatty sequences, devious convergence, and partitional divergence</a>, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.

%H K. L. Kodandapani and S. C. Seth, <a href="/A005736/a005736.pdf">On combinational networks with restricted fan-out</a>, IEEE Trans. Computers, 27 (1978), 309-318. (Annotated scanned copy)

%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrab.html">The Fibonacci Rabbit Sequence</a>

%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1810.09787">The Tribonacci and ABC Representations of Numbers are Equivalent</a>, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

%H G. Melancon, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00123-5">Lyndon factorization of sturmian words</a>, Discr. Math., 210 (2000), 137-149.

%H S. Mneimneh, <a href="https://doi.org/10.1145/2676723.2677215">Fibonacci in The Curriculum: Not Just a Bad Recurrence</a>, in Proceeding SIGCSE '15 Proceedings of the 46th ACM Technical Symposium on Computer Science Education, 253-258.

%H C. Mongoven, <a href="http://caseymongoven.com/b35">The Rabbit Sequence</a> (a musical composition with A005614).

%H T. D. Noe, <a href="/A171676/a171676.txt">The first 1652 subwords, including leading zeros</a>.

%H José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, <a href="http://arxiv.org/abs/1212.1368">A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake</a>, arXiv preprint arXiv:1212.1368 [cs.DM], 2012-2014.

%H Jeffrey Shallit, <a href="http://cs.uwaterloo.ca/~shallit/papers.html">Characteristic words as fixed points of homomorphisms</a>, University of Waterloo Technical Report CS-91-72, 1991.

%H Jeffrey Shallit, <a href="/A005614/a005614.ps">Characteristic words as fixed points of homomorphisms</a>. [Cached copy, with permission]

%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)

%H K. B. Stolarsky, <a href="http://dx.doi.org/10.4153/CMB-1976-071-6">Beatty sequences, continued fractions and certain shift operators</a>, Canad. Math. Bull., 19 (1976), 473-482.

%H Scott V. Tezlaf, <a href="https://arxiv.org/abs/1806.00331">On ordinal dynamics and the multiplicity of transfinite cardinality</a>, arXiv:1806.00331 [math.NT], 2018. See p. 10.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RabbitConstant.html">Rabbit Constant</a> and <a href="http://mathworld.wolfram.com/RabbitSequence.html">Rabbit Sequence</a>.

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>

%F Define strings S(0)=1, S(1)=10, thereafter S(n)=S(n-1)S(n-2); iterate. Sequence is S(oo). The individual S(n)'s are given in A036299.

%F a(n) = floor((n+1)*u)-floor(n*u), where u = (-1+sqrt(5))/2.

%F Sum_{n>=0} a(n)/2^(n+1) = A014565. - _R. J. Mathar_, Jul 19 2013

%F From _Peter Bala_, Nov 11 2013: (Start)

%F If we read the present sequence as the digits of a decimal constant c = 0.101101011011010 ... then we have the series representation c = Sum_{n >= 1} 1/10^floor(n*phi). An alternative representation is c = Sum_{n >= 1} 1/10^floor(n/phi) - 10/9.

%F The constant 9*c has the simple continued fraction representation [0; 1, 10, 10, 100, 1000, ..., 10^Fibonacci(n), ...]. See A010100.

%F Using this result we can find the alternating series representation c = 1/9 - 9*Sum_{n >= 1} (-1)^(n+1)*(1 + 10^Fibonacci(3*n+1))/( (10^(Fibonacci(3*n - 1)) - 1)*(10^(Fibonacci(3*n + 2)) - 1) ). The series converges very rapidly: for example, the first 10 terms of the series give a value for c accurate to more than 5.7 million decimal places. Cf. A014565. (End)

%e The infinite word is 101101011011010110101101101011...

%p Digits := 50; u := evalf((1-sqrt(5))/2); A005614 := n->floor((n+1)*u)-floor(n*u);

%t Nest[ Flatten[ # /. {0 -> {1}, 1 -> {1, 0}}] &, {1}, 10] (* _Robert G. Wilson v_, Jan 30 2005 *)

%t Flatten[Nest[{#, #[[1]]} &, {1, 0}, 9]] (* _IWABUCHI Yu(u)ki_, Oct 23 2013 *)

%t SubstitutionSystem[{0 -> {1}, 1 -> {1, 0}}, {1, 0}, 9] // Last (* _Jean-François Alcover_, Feb 06 2020 *)

%o (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 for(n=2,10,print(n" "a(n,[ 0 ],[ 1 ],[ 1,0 ]))) \\ Gives successive convergents to sequence

%o (PARI) /* for m>=1 compute exactly A183136(m+1)+1 terms of the sequence */

%o r=(1+sqrt(5))/2;v=[1,0];for(n=2,m,v=concat(v,vector(floor((n+1)/r),i,v[i]));a(n)=v[n];) /* _Benoit Cloitre_, Jan 16 2013 */

%o a005614 n = a005614_list !! n

%o a005614_list = map (1 -) a003849_list

%o -- _Reinhard Zumkeller_, Apr 07 2012

%o (MAGMA) [Floor((n+1)*(-1+Sqrt(5))/2)-Floor(n*(-1+Sqrt(5))/2): n in [1..100]]; // _Vincenzo Librandi_, Jan 17 2019

%Y Binary complement of A003849, which is the standard form of this sequence.

%Y Two other essentially identical sequences are A096270, A114986.

%Y Subwords: A178992, A171676.

%Y Cf. A000045 (Fibonacci numbers), A001468, A001911, A005206 (partial sums), A014565, A014675, A022342, A036299, A044432, A221150, A221151, A221152.

%Y The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - _N. J. A. Sloane_, Mar 11 2021

%K nonn,easy,nice

%O 0,1

%A _N. J. A. Sloane_

%E Corrected by _Clark Kimberling_, Oct 04 2000

%E Name corrected by _Michel Dekking_, Apr 02 2019

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Last modified April 10 07:44 EDT 2021. Contains 342843 sequences. (Running on oeis4.)