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Search: id:A116178
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| A116178 |
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Stewart's choral sequence. |
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+0
4
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| 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
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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The sequence is cube-free, i.e. it contains no substrings of the form XXX where X is a sequence of 0s and 1s.
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. [From 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.). [From Joel Reyes Noche (joel.noche(AT)up.edu.ph), Nov 01 2009]
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REFERENCES
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Ian Stewart, How to Cut a Cake and Other Mathematical Conudrums, Chapter 6.
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]
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FORMULA
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a(3n) = 0, a(3n-1) = 1 and a(3n+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
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PROGRAM
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(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 */
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CROSSREFS
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Cf. A010060.
Sequence in context: A122414 A144605 A154103 this_sequence A028999 A091244 A131378
Adjacent sequences: A116175 A116176 A116177 this_sequence A116179 A116180 A116181
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KEYWORD
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easy,nonn
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AUTHOR
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Richard Forster (gbrl01(AT)yahoo.co.uk), Apr 15 2007
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