A116178 Stewart's choral sequence: a(3n) = 0, a(3n-1) = 1, a(3n+1) = a(n).

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

Offset: 0

Author

Richard Forster (gbrl01(AT)yahoo.co.uk), Apr 15 2007

Keywords

• easy
• nonn

Comments

The sequence is cubefree, i.e., it contains no substrings of the form XXX where X is a sequence of 0's and 1's.

The sequence is the same as the infinite binary word w(infty) generated by w(n+1)=w(n)w(n)w*(n), where n is in {0,1,2,...}, w(0)=0 and w*(n) is w(n) with the middle letter changed. (Example: w*(0)=1, w(1)=001, w*(1)=011, w(2)=001001011.) - Joel Reyes Noche (joel.noche(AT)up.edu.ph), Mar 24 2008

The sequence is the fixed point of the morphism 0->001, 1->011, starting from a(0) = 0. - Joel Reyes Noche (joel.noche(AT)up.edu.ph), Apr 28 2008

A generalized choral sequence c(3n+r_0)=0, c(3n+r_1)=1, c(3n+r_c)=c(n), with r_0=0, r_1=2, and r_c=1. - Joel Reyes Noche (joel.noche(AT)up.edu.ph), Jul 09 2009

It is an infinite Lyndon word; it has an infinite number of prefixes which are Lyndon words (0, 001, 001001011, etc.). - Joel Reyes Noche (joel.noche(AT)up.edu.ph), Nov 01 2009

This sequence (with offset 1) is given by a(3k-2)=0, a(3k-1)=a(k), a(3k)=1-a(k) for k>=1, a(0)=0; for sequences generated by such recurrences, see A189628. - Clark Kimberling, Apr 28 2011

Van der Waerden's theorem tells us there can be no infinite binary word avoiding a monochromatic arithmetic progression of length 5 (the longest is of length 177; see A121894). However, Stewart's choral sequence has the property that it has no ababa appearing in arithmetic progression, for a different from b. - Jeffrey Shallit, Jul 03 2020

References

• J.R. Noche, On Stewart's Choral Sequence, Gibon, 8 (2008), 1-5. [From Joel Reyes Noche (joel.noche(AT)up.edu.ph), Aug 20 2008]
• J. R. Noche, Generalized Choral Sequences, Matimyas Matematika, 31 (2008), 25-28. [From Joel Reyes Noche (joel.noche(AT)up.edu.ph), Jul 09 2009]
• Ian Stewart, How to Cut a Cake and Other Mathematical Conundrums, Chapter 6.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..19683
• F. M. Dekking, Permutations of N generated by left-right filling algorithms, arXiv:2001.08915 [math.CO], 2020.
• Gabriele Fici and Jeffrey Shallit, Properties of a Class of Toeplitz Words, arXiv:2112.12125 [cs.FL], 2021.
• Robbert Fokkink and Gandhar Joshi, Anti-recurrence sequences, arXiv:2506.13337 [math.NT], 2025. See pp. 12, 18.

Crossrefs

Cf. A010060, A189636, A189637, A189638, A121894.

Programs

Mathematica

t = Nest[Flatten[# /. {0->{0,0,1}, 1->{0,1,1}}] &, {0}, 5] (*A116178*)
f[n_] := t[[n]]
Flatten[Position[t, 0]] (*A189636*)
Flatten[Position[t, 1]] (*A189637*)
s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
Table[s[n], {n, 1, 120}] (*A189638*)
(* Clark Kimberling, Apr 24 2011 *)

PARI

{a(n)= if(n<0, 1-a(-1-n), if(n%3==0, 0, if(n%3==2, 1, a(n\3))))} /* Michael Somos, Apr 17 2007 */

Formula

a(3*n) = 0, a(3*n-1) = 1 and a(3*n+1) = a(n).

G.f.: x^2/(1-x^3) +x^7/(1-x^9) +x^22/(1-x^27) +... . a(-1-n) = 1-a(n). - Michael Somos, Apr 17 2007

a(k)=1 if k=3^{m+1}n+(1/2)(5*3^m-1) and a(k)=0 if k=3^{m+1}n+(1/2)(3^m-1) for m,n in {0,1,2,...}. - Joel Reyes Noche (joel.noche(AT)up.edu.ph), Mar 24 2008

Extensions

Formula added to the name by Antti Karttunen, Aug 31 2017