A092782 The ternary tribonacci word; also a Rauzy fractal sequence: fixed point of the morphism 1 -> 12, 2 -> 13, 3 -> 1, starting from a(1) = 1.

1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3

Offset

1,2

Comments

See A080843 for the {0,1,2} version, which in a sense is the most basic version.

See also A103269 for another version with further references and comments.

Also called a tribonacci word. In the limit the ratios #1's : #2's : #3's are t^2 : t : 1 where t is the tribonacci constant 1.839286755... (A058265). - Frank M Jackson, Mar 29 2018

a(n)-1 is the number of trailing 0's in the maximal tribonacci representation of n (A352103). - Amiram Eldar, Feb 29 2024

References

This entry has a fairly complete list of references and links concerning the ternary tribonacci word. - N. J. A. Sloane, Aug 17 2018

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 246.

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..19513

Pierre Arnoux and Edmund Harriss, What is a Rauzy Fractal?, Notices Amer. Math. Soc., 61 (No. 7, 2014), 768-770, also p. 704 and front cover.

Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.

Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320. See T on page 315.

Marcy Barge and Jaroslaw Kwapisz, Geometric theory of unimodular Pisot substitutions, Amer. J. Math. 128 (2006), no. 5, 1219--1282. MR2262174 (2007m:37039). See Fig. 18.1. - N. J. A. Sloane, Aug 06 2014

Jean Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull. EATCS, #79 (2003), pp. 178-228.

Nataliya Chekhova, Pascal Hubert, and Ali Messaoudi, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, Journal de théorie des nombres de Bordeaux, 13.2 (2001): 371-394.

David Damanik and Luca Q. Zamboni, Arnoux-Rauzy subshifts: linear recurrence, powers and palindromes, arXiv:math/0208137 [math.CO], 2002.

Aldo de Luca and Luca Q. Zamboni, On prefixal factorizations of words, arXiv:1505.02309 [math.CO], 2015. See Example 2.

Aldo de Luca and Luca Q. Zamboni, On prefixal factorizations of words, European Journal of Combinatorics, Volume 52, Part A, 2016, pp. 59-73. See Example 2.

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Eric Duchêne and Michel Rigo, A morphic approach to combinatorial games: the Tribonacci case. RAIRO - Theoretical Informatics and Applications, 42, 2008, pp 375-393. doi:10.1051/ita:2007039. [Also available here]

Jacques Justin and Laurent Vuillon, Return words in Sturmian and episturmian words, RAIRO-Theoretical Informatics and Applications 34.5 (2000): 343-356. See Example on page 345.

Aayush Rajasekaran, Narad Rampersad, and Jeffrey Shallit, Overpals, Underlaps, and Underpals, In: Brlek S., Dolce F., Reutenauer C., Vandomme É. (eds) Combinatorics on Words, WORDS 2017, Lecture Notes in Computer Science, vol 10432.

Gérard Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110.2 (1982): 147-178. See page 148.

Victor F. Sirvent, Semigroups and the self-similar structure of the flipped tribonacci substitution, Applied Math. Letters, 12 (1999), 25-29. [Contains further related references.]

Victor F. Sirvent, The common dynamics of the Tribonacci substitutions, Bulletin of the Belgian Mathematical Society-Simon Stevin 7.4 (2000): 571-582.

Bo Tan and Zhi-Ying Wen, Some properties of the Tribonacci sequence, European Journal of Combinatorics, 28 (2007) 1703-1719.

Ondřej Turek, Abelian Complexity Function of the Tribonacci Word, J. Int. Seq. 18 (2015) Article 15.3.4.

Index entries for sequences that are fixed points of mappings.

Formula

a(n) = 1 for n in A003144; a(n) = 2 for n in A003145; a(n) = 3 for n in A003146.

a(n) = A080843(n-1) + 1. - Joerg Arndt, Sep 14 2013

Example

From Joerg Arndt, Sep 14 2013: (Start)
The first few steps of the substitution are
Start: 1
Maps:
  1 --> 12
  2 --> 13
  3 --> 1
-------------
0:   (#=1)
  1
1:   (#=2)
  12
2:   (#=4)
  1213
3:   (#=7)
  1213121
4:   (#=13)
  1213121121312
5:   (#=24)
  121312112131212131211213
6:   (#=44)
  12131211213121213121121312131211213121213121
7:   (#=81)
  121312112131212131211213121312112131212131211213121121312121312112131213121121312
(End)

Maple

f(1):= (1, 2): f(2):= (1, 3): f(3):= (1): A:= [1]:
for i from 1 to 16 do A:= map(f, A) od:
A; # 19513 terms of A092782; A103269; from N. J. A. Sloane, Aug 06 2018

Mathematica

Nest[ Flatten[# /. {1 -> {1, 2}, 2 -> {1, 3}, 3 -> 1}] &, {1}, 8] (* Robert G. Wilson v, Mar 04 2005 and updated Apr 29 2018 *)

Prog

(PARI) w=vector(9, x, []); w[1]=[1];
for(n=2, 9, for(k=1, #w[n-1], m=w[n-1][k]; v=[]; if(m-1, if(m-2, v=[1], v=[1, 3]), v=[1, 2]); w[n]=concat(w[n], v)));
w[9] \\ Gerald McGarvey, Dec 18 2009
(PARI)
strsub(s, vv, off=0)=
{
    my( nl=#vv, r=[], ct=1 );
    while ( ct <= #s,
        r = concat(r, vv[ s[ct] + (1-off) ] );
        ct += 1;
    );
    return( r );
}
t=[1];  for (k=1, 10, t=strsub( t, [[1, 2], [1, 3], [1]], 1 ) );  t
\\ Joerg Arndt, Sep 14 2013
(PARI) A092782_vec(N, s=[[1, 2], [1, 3], 1], A=[1])={while(#A<N, A=concat(vecextract(s, A))); A} \\ Return at least N terms. - M. F. Hasler, Dec 14 2018

Crossrefs

Cf. A003144, A003145, A003146, A100619, A103269, A073058, A245553, A245554, A105083, A352103.

See A080843 for a {0,1,2} version.

First differences: A317950.

Sequence in context: A279288 A078734 A028293 * A119647 A228098 A366833

Adjacent sequences: A092779 A092780 A092781 * A092783 A092784 A092785

Keyword

easy,nonn

Author

Philippe Deléham, Apr 23 2004

Extensions

Additional references and links added by N. J. A. Sloane, Aug 17 2018

Status

approved