A049321 -id:A049321 - cp's OEIS frontend

A049320 Non-primitive Chacon sequence: fixed under 0->0010, 1->1.

0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0

Offset: 1

N. J. A. Sloane

A word that is pure morphic and primitive morphic, but neither uniform morphic nor pure primitive morphic. - N. J. A. Sloane, Jul 14 2018
This is A133162 on the alphabet {0,1}, instead of {1,2}. - Michel Dekking, Oct 24 2019
The [10->1]-transform of (a(n)) is the sequence A189640. - Michel Dekking, Oct 26 2019

Reinhard Zumkeller, Table of n, a(n) for n = 1..10001 [indexing adapted by _Georg Fischer_, Oct 25 2019]
J.-P. Allouche, M. Baake, J. Cassaigns, and D. Damanik, Palindrome complexity, arXiv:math/0106121 [math.CO], 2001; Theoretical Computer Science, 292 (2003), 9-31.
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
R. V. Chacon, Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc., 22 (1969), pp. 559-562.
Fabien Durand, Julien Leroy, and Gwenaël Richomme, Do the Properties of an S-adic Representation Determine Factor Complexity?, Journal of Integer Sequences, Vol. 16 (2013), #13.2.6.
S. Ferenczi, Complexity of sequences and dynamical systems, Discrete Math., 206 (1999), 145-154.
Konstantinos Karamanos, Entropy analysis of substitutive sequences revisited, Journal of Physics A: Mathematical and General 34.43 (2001): pages 9231-9241. See Eq. (31).

Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.

Cf. A003849, A049321, A133162, A317962.

a049320 n = a049320_list !! n
a049320_list = 0 : 0 : 1 : 0 : f [0,0,1,0] where
   f xs = drop (length xs) ys ++ f ys where
     ys = concatMap ch xs
     ch 0 = [0,0,1,0]; ch 1 = [1]
-- Reinhard Zumkeller, Aug 14 2013

Nest[# /. 0 -> {0, 0, 1, 0}&, {0}, 4] // Flatten (* Jean-François Alcover, Oct 08 2016 *)

Offset changed by Michel Dekking, Oct 24 2019

A131989 Start with the symbol **|* and for each iteration replace * with **|. This sequence is the number of 's between each dash.

Original entry on oeis.org

2, 3, 1, 2, 3, 3, 1, 2, 1, 2, 3, 1, 2, 3, 3, 1, 2, 3, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 3, 1, 2, 1, 2, 3, 1, 2, 3, 3, 1, 2, 3, 3, 1, 2, 1, 2, 3, 1, 2, 3, 3, 1, 2, 3, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 3, 1, 2, 1, 2, 3, 1, 2, 3, 3, 1, 2, 3

Offset: 1

Alex H. Bishop (AlexanderBishop(AT)stmarksschool.org), Oct 07 2007

If the leading a(1)=2 is dropped, at least the next 90 terms coincide with those of A026181. - R. J. Mathar, Jun 13 2008
From Michel Dekking, Oct 19 2019: (Start)
Coding * by 1 and | by 2, the procedure is the same as performing the substitution sigma: 1 -> 1121, 2 -> 2. The return words of this morphism are 12, 112 and 1112. Under sigma these words transform as
      12->11212, 112-> 112111212, 1112->1121112111212.
Coding the return words by their length minus one, the corresponding derivated morphism is
      1-> 21, 2-> 231, 3-> 2331.
(a(n)) is the unique fixed point of this morphism.
(End)
a(n) = x(n+1), where x is the primitive Chacon sequence A049321 on the alphabet {3,1,2} instead of {0,1,2}. This follows from the fact that
  sigma: 0->0012,  1->12, 2->012  and  tau: 0->2001, 1->21, 2->201 are conjugated morphisms: tau(j) = 2 sigma(j) 2^{-1} for j=0,1,2. - Michel Dekking, Oct 23 2019

			The symbol through a few iterations: **|*, **|***|*|**|*, **|***|*|**|***|***|*|**|*|**|***|*|**|*, etc.

a(n) = length of n-th run of 1's in A133162. - N. J. A. Sloane, Oct 09 2007

Comments from N. J. A. Sloane, Oct 10 2007: (Start)
The following is a simple recursive method to generate this sequence. The sequence is lim_{ t -> oo } S_t, where S_0 = 1+, and S_{t+1} is obtained from the concatenation S_t S_t S_t by replacing the first + by the sum of the two numbers adjacent to it and deleting the second +.
Thus we have:
S_0 = 1+,
S_1 = 1+1+1+ -> 21+,
S_2 = 21+21+21+ -> 23121+,
S_3 = 23121+23121+23121+ -> 23123312123121+,
S_4 = 23123312123121+23123312123121+23123312123121+ -> 23123312123123312331212312123123312123121+, etc.
Denote the sequence by a(1), a(2), ...
Block t, that is, S_t omitting the final 1+, extends from n=1 through n=(3^t-1)/2.
Given n, to find a(n): first find t from
p = (3^(t-1)-1)/2 < n <= (3^t-1)/2.
Assume t >= 2. Then if n=(3^(t-1)+1)/2, a(n) = 3 and if n=3^(t-1), a(n) = 1.
Otherwise, a(n) = a(n'), where
n' = n-p if n<3^(t-1), otherwise n' = n-3^(t-1). (End)

More terms from N. J. A. Sloane, Oct 10 2007

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A049320 Non-primitive Chacon sequence: fixed under 0->0010, 1->1.

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A131989 Start with the symbol **|* and for each iteration replace * with **|. This sequence is the number of 's between each dash.

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cp's OEIS Frontend

A049320 Non-primitive Chacon sequence: fixed under 0->0010, 1->1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A131989 Start with the symbol **|* and for each iteration replace * with **|*. This sequence is the number of *'s between each dash.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A131989 Start with the symbol **|* and for each iteration replace * with **|. This sequence is the number of 's between each dash.