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A284775 Fixed point of the morphism 0 -> 01, 1 -> 0011. 11
0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
COMMENTS
Let u(n) = # 0's <= n and v(n) = # 1's <= n. Then 2n - u(n) is in {1,2} and 2n-v(n) is in {-1,0}, for all n>=1.
This conjecture is false, since 2n - u(n) > n. - Michel Dekking, Oct 14 2019
From Michel Dekking, Oct 14 2019: (Start)
This sequence, as a word, has the remarkable property that it is an automatic sequence, i.e., the letter-to-letter image of a fixed point of a uniform morphism, given by
a->abc, b->deb, c->aba, d->bcd, e->ebc.
The letter-to-letter map is given by a->0, b-> 1, c->0, d->0, e->1.
For an algorithm to find this morphism, and the letter-to-letter map see Section V of the paper "The spectrum of dynamical systems arising from substitutions of constant length".
(N.B. The algorithm gives a morphism on an alphabet of 6 letters, but it is possible to merge two of them, because they have the same images under the morphism and under the letter-to-letter map.)
(End)
LINKS
J.-P. Allouche, F. M. Dekking, and M. Queffélec, Hidden automatic sequences, arXiv:2010.00920 [math.NT], 2020.
F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Publications mathématiques et informatique de Rennes, no. 2 (1976), Exposé no. 6, 34 p.
F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und verw. Gebiete 41 (1978), 221-239.
FORMULA
a(n) = 1 if (a(n-2)=0, a(n-1)=0) or (a(n-3)=0, a(n-2)=1, a(n-1)=0); otherwise a(n)=0.
This formula is not correct, since a(9)=0, a(10)=1, a(11)=0, but a(12)=0. - Michel Dekking, Oct 14 2019
EXAMPLE
0 -> 01-> 0011 -> 010100110011 ->
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 0, 1, 1}}] &, {0}, 7]; (* A284775 *)
Flatten[Position[s, 0]]; (* A284776 *)
Flatten[Position[s, 1]]; (* A284777 *)
Flatten[SubstitutionSystem[{0->{0, 1}, 1->{0, 0, 1, 1}}, {0}, {5}]] (* Harvey P. Dale, Jun 18 2022 *)
CROSSREFS
Cf. A284776, A284777, A284793 (difference sequence; 3 distinct terms), A284794, A284795, A284796.
Sequence in context: A059448 A283318 A288633 * A156259 A038219 A330731
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 14 2017
STATUS
approved

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Last modified June 16 07:19 EDT 2024. Contains 373423 sequences. (Running on oeis4.)