A039982 Let phi denote the morphism 0 -> 11, 1 -> 10. This sequence is the limit S(oo) where S(0) = 1; S(n+1) = 1.phi(S(n)).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

An example of a d-perfect sequence.

Concatenation of the bit sequences 1, 10, 1011, 10111010, 1011101010111011, ... used in a construction of A035263 (see Comment there by Benoit Cloitre). - David Callan, Oct 08 2005

Image, under the coding a,b,d -> 1, c -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cd, c -> cd, d -> bb. - Jeffrey Shallit, May 15 2016

Examples

			The first few S(i) are:
S(0) = 1
S(1) = 1.10 = 110
S(2) = 1.101011 = 1101011
S(3) = 1.10101110111010 = 110101110111010
...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• Martin Klazar and Florian Luca, On integrality and periodicity of the Motzkin numbers.
• Martin Klazar and Florian Luca, On integrality and periodicity of the Motzkin numbers, Aequationes Math. 69 (2005), no. 1-2, 68-75.
• D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions
• D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, Sequences and their Applications, Discrete Mathematics and Theoretical Computer Science 1999, pp 308-317.
• Index entries for characteristic functions

Crossrefs

Cf. A001006, A035263, A090344, A091090.

Programs

GAP

b:=[1,1,2];; for n in [4..120] do b[n]:=(1/(n+1))* (2*n*b[n-1]+(3*n-7)*b[n-2]-(4*n-10)*b[n-3]);; od; a:=b mod 2; # Muniru A Asiru, Sep 28 2018

Mathematica

substitutionRule={1->{1, 0}, 0->{1, 1}}; makeSubstitution[seq_]:=Flatten[seq/.substitutionRule]; Flatten[NestList[makeSubstitution, {1}, 5]]
NestList[Flatten[ # /. {0 -> {1, 1}, 1 -> {1, 0}}] &, {1}, 6] // Flatten (* Robert G. Wilson v, Mar 29 2006 *)

PARI

a(n)=my(A=1+x); for(i=1, n, A=1/(1-x+x*O(x^n))+x^2*A^2+x*O(x^n)); polcoeff(A, n)%2 \\ Charles R Greathouse IV, Feb 04 2013

PARI

up_to = 16384;
A090344list(up_to) = { my(v=vector(up_to)); v[1] = 1; v[2] = 2; v[3] = 3; for(n=4,up_to,v[n] = ((2*n+2)*v[n-1] -(4*n-6)*v[n-3] +(3*n-4)*v[n-2])/(n+2)); (v); };
v090344 = A090344list(up_to);
A090344(n) = if(!n,1,v090344[n]);
A039982(n) = (A090344(n)%2); \\ Antti Karttunen, Sep 27 2018

Formula

a(n) = A090344(n) mod 2. - Christian G. Bower, Jun 12 2005

a(n) = A091090(n+1) mod 2. - Alan Michael Gómez Calderón, Jul 05 2025

Extensions

More terms from Christian G. Bower, Jun 12 2005

Offset corrected from 1 to 0 by Antti Karttunen, Sep 27 2018

Entry revised by N. J. A. Sloane, Feb 23 2019