A284775 Fixed point of the morphism 0 -> 01, 1 -> 0011.

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

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

Clark Kimberling, Table of n, a(n) for n = 1..10000

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.

Index entries for sequences that are fixed points of mappings

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

Adjacent sequences: A284772 A284773 A284774 * A284776 A284777 A284778

Keyword

nonn,easy

Author

Clark Kimberling, Apr 14 2017

Status

approved