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A036577 Ternary Thue-Morse sequence: closed under a->abc, b->ac, c->b. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

REFERENCES

Currie, James D. "Non-repetitive words: Ages and essences." Combinatorica 16.1 (1996): 19-40. See p. 21.

M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 26.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

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.

J. Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.

David Hamm, and Jeffrey Shallit, Characterization of finite and one-sided infinite fixed points of morphisms on free monoids, Technical Report CS-99-17, Dep. of Computer Science, University of Waterloo, 1999. See foot of page 2.

Michaël Rao, Michel Rigo, Pavel Salimov, Avoiding 2-binomial squares and cubes, arXiv:1310.4743 [cs.FL], 2013.

Michaël Rao, Michel Rigo, Pavel Salimov, Avoiding 2-binomial squares and cubes, Theoretical Computer Science, Volume 572, 23 March 2015, Pages 83-91.

FORMULA

a(n) = A036585(n) - 1 = A029883(n) + 1.

a(n) = 3 - A007413(n). a(A036554(n)) = 1; a(A091785(n)) = 0; a(A091855(n)) = 2. - Philippe Deléham, Mar 20 2004

a(4*n + 2) = 1. a(2*n + 1) = 2 * A010059(n). a(4*n + 3) = 2 * A010060(n). - Michael Somos, Aug 03 2011

EXAMPLE

2*x + x^2 + 2*x^4 + x^6 + 2*x^7 + x^8 + x^10 + 2*x^11 + 2*x^13 + x^14 + ...

MATHEMATICA

(* 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 *)

PROG

(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 */

CROSSREFS

Cf. A010059, A010060, A029883, A036585, A104248.

See A007413, A036580 for other versions.

Sequence in context: A298731 A321102 A091392 * A317189 A220136 A322454

Adjacent sequences:  A036574 A036575 A036576 * A036578 A036579 A036580

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 20 17:05 EST 2019. Contains 319335 sequences. (Running on oeis4.)