

A005614


The infinite Fibonacci word (start with 1, apply 0>1, 1>10, iterate, take limit).


59



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, 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, 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
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OFFSET

0,1


COMMENTS

Characteristic function of A022342.  Philippe Deléham, May 03 2004
a(n) = number of 0's between successive 1's (see also A003589 and A007538).  Eric Angelini, Jul 06 2005
With offset 1 this is the characteristic sequence for Wythoff Anumbers A000201=[1,3,4,6,...].
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
For generalized Fibonacci words see A221150, A221151, A221152...  Peter Bala, Nov 11 2013
The limiting mean of the first n terms is phi  1; the limiting variance is phi (A001622).  Clark Kimberling, Mar 12 2014
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) by h, and apply the difference operator to every column. In the resulting array, every column is A005614.  Clark Kimberling, Mar 02 2015


REFERENCES

J.P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.
G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 3442, esp. p. 36.


LINKS

T. D. Noe, Table of n, a(n) for n=0..10945 (19 iterations)
F. Axel et al., Vibrational modes in a one dimensional "quasialloy": the Morse case, J. de Physique, Colloq. C3, Supp. to No. 7, Vol. 47 (Jul 1986), pp. C3181C3186; see Eq. (10).
Joerg Arndt, Matters Computational (The Fxtbook), pp.753754
M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139151.
S. Dulucq and D. GouyouBeauchamps, Sur les facteurs des suites de Sturm, Theoret. Comput. Sci. 71 (1990), 381400.
M. S. El Naschie, Statistical geometry of a Cantor discretum and semiconductors, Computers & Mathematics with Applications, Vol. 29 (Issue 12, June 1995), 103110.
D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 3543.
J. Grytczuk, Infinite semisimilar words, Discrete Math. 161 (1996), 133141.
C. Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267273.
Ron Knott, The Fibonacci Rabbit Sequence
G. Melancon, Lyndon factorization of sturmian words, Discr. Math., 210 (2000), 137149.
S. Mneimneh, Fibonacci in The Curriculum: Not Just a Bad Recurrence, in Proceeding SIGCSE '15 Proceedings of the 46th ACM Technical Symposium on Computer Science Education, Pages 253258.
C. Mongoven, The Rabbit Sequence (a musical composition with A005614)
T. D. Noe, The first 1652 subwords, including leading zeros.
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
Jeffrey Shallit, Characteristic words as fixed points of homomorphisms, University of Waterloo Technical Report CS9172, 1991.
Jeffrey Shallit, Characteristic words as fixed points of homomorphisms. [Cached copy, with permission]
K. B. Stolarsky, Beatty sequences, continued fractions and certain shift operators, Canad. Math. Bull., 19 (1976), 473482.
Eric Weisstein's World of Mathematics, Rabbit Constant
Eric Weisstein's World of Mathematics, Rabbit Sequence
Index entries for characteristic functions
Index entries for sequences that are fixed points of mappings


FORMULA

Define strings S(0)=1, S(1)=10, thereafter S(n)=S(n1)S(n2); iterate. Sequence is S(oo). The individual S(n)'s are given in A036299.
a(n) = floor((n+1)*u)floor(n*u), where u = (1+sqrt(5))/2.
Sum_{n>=0} a(n)/2^(n+1) = A014565.  R. J. Mathar, Jul 19 2013
From Peter Bala, Nov 11 2013: (Start)
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.
The constant 9*c has the simple continued fraction representation [0; 1, 10, 10, 100, 1000, ..., 10^Fibonacci(n), ...]. See A010100.
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)


EXAMPLE

The infinite word is 101101011011010110101101101011...


MAPLE

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


MATHEMATICA

Nest[ Flatten[ # /. {0 > {1}, 1 > {1, 0}}] &, {1}, 10] (* Robert G. Wilson v, Jan 30 2005 *)
Flatten[Nest[{#, #[[1]]} &, {1, 0}, 9]] (* IWABUCHI Yu(u)ki, Oct 23 2013 *)


PROG

(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
for(n=2, 10, print(n" "a(n, [ 0 ], [ 1 ], [ 1, 0 ]))) \\ Gives successive convergents to sequence
(PARI) /* for m>=1 compute exactly A183136(m+1)+1 terms of the sequence */
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 */
(Haskell)
a005614 n = a005614_list !! n
a005614_list = map (1 ) a003849_list
 Reinhard Zumkeller, Apr 07 2012


CROSSREFS

Binary complement of A003849, which is the standard form of this sequence. Cf. A036299, A001468, A014675.
Two other essentially identical sequences are A096270, A114986.
Subwords: A178992, A171676.
Cf. A000045 (Fibonacci numbers), A044432, A005206 (partial sums). A014565, A221150, A221151, A221152, A001911, A022342.
Sequence in context: A096055 A125144 A115198 * A272532 A166946 A174208
Adjacent sequences: A005611 A005612 A005613 * A005615 A005616 A005617


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Corrected by Clark Kimberling, Oct 04 2000


STATUS

approved



