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A036577
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Ternary Thue-Morse sequence: closed under a->abc, b->ac, c->b.
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18
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2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0
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OFFSET
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1,1
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COMMENTS
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Number of 1's between successive 0's in A010060.
The infinite sequence is abcacbabcbac... which is encoded with a->2, b->1, c->0 to produce this integer sequence.
This word is sometimes called 'vtm'; see, for example, see the Blanchet-Sadri et al. reference.
It is a squarefree word containing no instances of the factor 010 or 212 (or cbc, aba in the encoding).
Berstel proved many different definitions (e.g., Braunholtz, Istrail) of the word are equivalent. (End)
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REFERENCES
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M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 26.
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LINKS
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J.-P. Allouche and Jeffrey Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
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FORMULA
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EXAMPLE
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2*x + x^2 + 2*x^4 + x^6 + 2*x^7 + x^8 + x^10 + 2*x^11 + 2*x^13 + x^14 + ...
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MATHEMATICA
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(* ThueMorse is built-in since version 10.2, for lower versions it needs to be defined manually *) ThueMorse[n_] := Mod[DigitCount[n, 2, 1], 2]; Table[1 + ThueMorse[n] - ThueMorse[n-1], {n, 1, 100}] (* Vladimir Reshetnikov, May 17 2016 *)
Nest[Flatten[# /. {2 -> {2, 1, 0}, 1 -> {2, 0}, 0 -> {1}}] &, {2, 1, 0}, 7] (* Robert G. Wilson v, Jul 30 2018 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, if( valuation( n, 2)%2, 1, 1 - (-1)^subst( Pol( binary(n)), x, 1)))} /* Michael Somos, Aug 03 2011 */
(Python)
def A036577(n): return (n.bit_count()&1)+((n-1).bit_count()&1^1) # Chai Wah Wu, Mar 03 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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