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A007413 A squarefree (or Thue-Morse) ternary sequence: closed under 1->123, 2->13, 3->2. Start with 1.
(Formerly M0406)
20
1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n)=2 if and only if n-1 is in A079523. - Benoit Cloitre, Mar 10 2003.

Partial sums modulo 4 of the sequence 1, a(1), a(1), a(2), a(2), a(3), a(3), a(4), a(4), a(5), a(5), a(6), a(6), ...- Philippe Deléham, Mar 04 2004

To construct the sequence : start with 1 and concatenate 4 -1 = 3 : 1, 3, then change the last term (2 -> 1, 3 ->2 ) gives : 1, 2. Concatenate 1, 2 with 4 -1 = 3, 4 - 2 = 2 : 1, 2, 3, 2 and change the last term : 1, 2, 3, 1. Concatenate 1, 2, 3, 1 with 4 - 1 = 3, 4 - 2 = 2, 4 - 3 = 1, 4 - 1 = 3 : 1, 2, 3, 1, 3, 2, 1, 3 and change the last term : 1, 2, 3, 1, 3, 2, 1, 2 etc.- Philippe Deléham, Mar 04 2004

To construct the sequence : start with the Thue-Morse sequence A010060 = 0, 1, 1, 0, 1, 0, 0, 1, ... Then change 0 -> 1, 2, 3, _ and 1 -> 3, 2, 1, _ gives : 1, 2, 3, _, 3, 2, 1, _,3, 2, 1, _, 1, 2, 3, _, 3, 2, 1, _, ...and fill in the successive holes with the successive terms of the sequence itself.- Philippe Deléham, Mar 04 2004

To construct the sequence : to insert the number 2 between the A003156(k)-th term and the (1 + A003156(k))-th term of the sequence 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, ...- Philippe Deléham, Mar 04 2004

Conjecture. The sequence is formed by the numbers of 1's between every pair of consecutive 2's in A076826. - Vladimir Shevelev, May 31 2009

REFERENCES

James D. Currie, Palindrome positions in ternary square-free words, Theoretical Computer Science, 396 (2008) 254-257

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 18.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. Thue. Über unendliche Zeichenreihe. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, 7:1a22, 1906.

LINKS

Table of n, a(n) for n=1..105.

Roger L. Bagula, Description of sequence as successive rows of a triangle

V. Keranen, New Abelian Square-Free DT0L-Languages over 4 Letters

S. Kitaev and T. Mansour, Counting the occurrences of generalized patterns....

FORMULA

a(n) modulo 2 = A035263(n). a(A036554(n)) = 2. a(A003159(n)) = 1 if n odd. a(A003159(n)) = 3 if n even. a(n) = A033485(n) mod 4. a(n) = 4 - A036585(n-1).- Philippe Deléham, Mar 04 2004

a(n) = 2 - A029883(n) = 3 - A036577(n) . - Philippe Deléham, Mar 20 2004

For n>=1, we have: 1) a(A108269(n))=A010684(n-1); 2) a(A079523(n))=A010684(n-1); 3) a(A081706(2n))=A010684(n). - Vladimir Shevelev, Jun 22 2009

EXAMPLE

Here are the first 5 stages in the construction of this sequence, together with Mma code, taken from Keranen's article. His alphabet is a,b,c rather than 1,2,3.

productions = {"a" → "abc ", "b" → "ac ", "c" → "b ", " " -> ""};

NestList[g, "a", 5] // TableForm

a

abc

abc ac b

abc ac b abc b ac

abc ac b abc b ac abc ac b ac abc b

abc ac b abc b ac abc ac b ac abc b abc ac b abc b ac abc b abc ac b ac

MATHEMATICA

Nest[ Flatten[ # /. {1 -> {1, 2, 3}, 2 -> {1, 3}, 3 -> {2}}] &, {1}, 7] (from Robert G. Wilson v, May 07 2005)

PROG

(PARI) a(n)=if(n<1|valuation(n, 2)%2, 2, 2+(-1)^subst(Pol(binary(n)), x, 1))

CROSSREFS

Cf. A001285, A010060.

First differences of A000069.

Equals A036580(n-1) + 1.

Cf. A115384 A159481 A007413 A000120.

Sequence in context: A128222 A057039 A135511 * A072457 A063047 A003270

Adjacent sequences:  A007410 A007411 A007412 * A007414 A007415 A007416

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 21 05:22 EST 2014. Contains 252296 sequences.