A317962 -id:A317962 - 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

A133162 Trajectory of 1 under the morphism 1 -> {1,1,2,1}, 2 -> {2}.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1

Offset: 1

N. J. A. Sloane, Oct 09 2007, Oct 10 2007

It can be shown that this is lim_{t -> oo} S_t, where S_0 = 1, S_{t+1} = S_t S_t 2 S_t.
Suggested by A131989: a(n) = length of n-th run of 1's in A131989.
For a proof of this see the Comments of A131989. - Michel Dekking, Oct 19 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences that are fixed points of mappings

Cf. A049320, A131989, A317962.

Nest[Function[l, {Flatten[(l /. {1 -> {1,1,2,1}, 2 -> {2} })] }], {1}, 5] (* Georg Fischer, Jul 19 2019 *)

Denote the sequence by a(1), a(2), ...
Block t, that is, S_t, extends from n=1 through n=(3^(t+1)-1)/2.
Given n, to find a(n): first find t from
p = (3^t-1)/2 < n <= (3^(t+1)-1)/2.
Then if n=3^t, a(n) = 2. Otherwise, a(n) = a(n'), where
n' = n-p if n<3^t, otherwise n' = n-2p-1.

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

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