A123740 Characteristic sequence for Wythoff AB-numbers A003623.

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

Offset: 1

Author

Wolfdieter Lang, Oct 13 2006

Keywords

• nonn
• easy

Comments

Left shifted sequence is the characteristic function of A035336, and also the second lowest digit of the Zeckendorf expansion of n. - Franklin T. Adams-Watters, Jun 30 2009

a(n) = A188009(n+2), n>=1. - Wolfdieter Lang, Jun 27 2011

Doubling the 0’s in the infinite Fibonacci word A003849 gives (a(n)). - Michel Dekking, Sep 09 2016

This is a morphic sequence, i.e., the letter-to-letter image of the fixed point of a morphism. The fixed point is the unique fixed point A270788 of the three symbol Fibonacci morphism. The letter-to-letter map is 1->0, 2->0, 3->1. - Michel Dekking, May 02 2019

References

• See references under A000201.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• Michel Dekking and Michael Keane, On the conjugacy class of the Fibonacci dynamical system, arXiv preprint arXiv:1608.04487 [math.DS], 2016.
• Michel Dekking and Michael Keane, On the conjugacy class of the Fibonacci dynamical system, Theoretical Computer Science 668 (2017), 59-69.
• Jeffrey Shallit and Anatoly Zavyalov, Transduction of Automatic Sequences and Applications, arXiv:2303.15203 [cs.FL], 2023, see p. 31.

Crossrefs

Cf. A003623, A000201, A001950, A188009, A270788.

Cf. A003849, A014417. - Franklin T. Adams-Watters, Jun 30 2009

Programs

Mathematica

a[] = 0; s = Table[n + 2 Floor[n*GoldenRatio], {n, 24}]; Map[Set[a[#], 1] &, s]; Array[a, Max[s]] (* _Michael De Vlieger, Mar 29 2023 *)

Python

from math import isqrt
def A123740(n): return (n+2+isqrt(m:=5*(n+2)**2)>>1)-(n+isqrt(m-20*(n+1))>>1)-3 # Chai Wah Wu, Aug 29 2022

Formula

a(n) = 1 if n=A(B(k)) for some k>=1, else 0, with A(k):=A000201(k) and B(k):=A001950(k), k>=1.

a(n) = 1-(1-h(n))-(1-h(n+1)) = h(n)-(1-h(n+1))= h(n)*h(n+1) with h(n):=A005614(n-1), n>=1, the rabbit sequence.

a(n) = A(n+2)-A(n)-3. - Wolfdieter Lang, Jun 27 2011