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A005614
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The infinite Fibonacci word (start with 1, apply 0->1, 1->10, iterate, take limit).
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51
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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
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0,1
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COMMENTS
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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 A-numbers 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). [From Remi Schulz (Remi.Schulz(AT)club-internet.fr), Jul 06 2010]
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.
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).
M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139-151.
S. Dulucq and D. Gouyou-Beauchamps, Sur les facteurs des suites de Sturm, Theoret. Comput. Sci. 71 (1990), 381-400.
M. S. El Naschie, Statistical geometry of a Cantor discretum and semiconductors, Computers Math. Applic., 29 (No, 12, 1995), 103-110.
J. Grytczuk, Infinite semi-similar words, Discrete Math. 161 (1996), 133-141.
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..10945 (19 iterations)
Joerg Arndt, Fxtbook, pp.753-754
R. Knott, The Fibonacci Rabbit Sequence
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
Eric Weisstein's World of Mathematics, Rabbit Constant
Eric Weisstein's World of Mathematics, Rabbit Sequence
Index entries for characteristic functions
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FORMULA
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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.
a(n) = floor((n+1)*u)-floor(n*u), where u = (1-sqrt(5))/2.
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EXAMPLE
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The infinite word is 101101011011010110101101101011...
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MAPLE
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Digits := 50; u := evalf((1-sqrt(5))/2); A005614 := n->floor((n+1)*u)-floor(n*u);
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MATHEMATICA
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Nest[ Flatten[ # /. {0 -> {1}, 1 -> {1, 0}}] &, {1}, 10] (* Robert G. Wilson v, Jan 30 2005*)
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PROG
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(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
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CROSSREFS
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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. A044432, A005206 (partial sums).
Sequence in context: A096055 A125144 A115198 * A166946 A174208 A141687
Adjacent sequences: A005611 A005612 A005613 * A005615 A005616 A005617
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Corrected by Clark Kimberling, Oct 04 2000
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STATUS
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approved
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