A245553 -id:A245553 - cp's OEIS frontend

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

Philippe Deléham, Apr 23 2004

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

			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)

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.

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.

Cf. A003144, A003145, A003146, A100619, A103269, A073058, A245553, A245554, A105083, A352103.
See A080843 for a {0,1,2} version.
First differences: A317950.

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

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

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

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

A092782_vec(N,s=[[1,2],[1,3],1],A=[1])={while(#AM. F. Hasler, Dec 14 2018

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

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

A073058 Define s(1)={1,2}, s(2)={1,3} and s(3)={1}. For a finite sequence A={a_1, ..., a_n}, with elements in {1,2,3}, define t(A) to be the concatenation of A, s(a_1), s(a_2), ... and s(a_n). Start with the sequence {1,2,3} and repeatedly apply t; limiting sequence is shown.

Original entry on oeis.org

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

Offset: 1

Roger L. Bagula, Aug 16 2002

nonn

A fractal sequence related to a sequence of Rauzy.

P. Arnoux and E. Harriss, What is a Rauzy Fractal?, Notices Amer. Math. Soc., 61 (No. 7, 2014), 768-770, also p. 704 and front cover.
Vincent Canterini and Anne Siegel, Geometric Representations of Substitutions of Pisot Type, Trans. Amer. Math. Soc. 353 (2001), 5121-5144.

Cf. A092782, A103684, A105083, A245553, A245554.

Nest[ Flatten[ Join[#, # /. {1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1}}]] &, {1, 2, 3},
4] (* Robert G. Wilson v, Jan 01 2017 *)

A105083 Trajectory of 1 under the morphism 1 -> 12, 2 -> 3, 3 -> 1.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Apr 06 2005

nonn

Aresh Pourkavoos, Table of n, a(n) for n = 0..10000
P. Arnoux and E. Harriss, What is a Rauzy Fractal?, Notices Amer. Math. Soc., 61 (No. 7, 2014), 768-770, also p. 704 and front cover.
Marcy Barge and Jaroslaw Kwapisz, Geometric theory of unimodular Pisot substitutions, Amer. J. Math. 128 (2006), no. 5, 1219--1282. MR2262174 (2007m:37039). - _N. J. A. Sloane_, Aug 06 2014
Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.
Victor F. Sirvent and Yang Wang, Self-Affine Tiling via Substitution Dynamical Systems and Rauzy Fractals, Pac. J. Math., 206 (2002), 465-485. See Example 2.2 and Figure 2 pp. 474-475.
Index entries for sequences that are fixed points of mappings

Cf. A005374, A073058, A092782, A202340, A245553, A245554, A000930.

Nest[ Function[ l, {Flatten[(l /. {1 -> {1, 2}, 2 -> {3}, 3 -> {1}})] }], {1}, 12]

N_TERMS=10000
def a():
  # Index of the current term
  n = 0
  # Stores the place values of the greedy representation of n,
  # minus two since A000930 begins with duplicate ones.
  places = []
  # Edge case: a(0)=1.
  yield 0, 1
  while True:
    n += 1
    # Add A000930(2+0)=1 to the representation of n
    places.append(0)
    # Apply carryover rule for as long as necessary:
    # if places contains n+2 and n,
    # both terms are replaced by n+3.
    while len(places) > 1 and places[-2] <= places[-1]+2:
      places.pop()
      places[-1] += 1
    # Look at the smallest term to decide a(n)
    an = 1 if places[-1] > 1 else places[-1]+2
    yield n, an
# Asymptotic behavior is O(log(n)*log(log(n))) memory
# and O(n) time to generate the first n terms,
# although a term may take as long as O(log(n)).
for n, an in a():
  print(n, an)
  if (n >= N_TERMS):
    break
# Aresh Pourkavoos, Jan 26 2021

From Aresh Pourkavoos, Jan 26 2021: (Start)
Limit S(infinity) of the following strings: S(0) = 2, S(1) = 1, S(2) = 0, S(n+3) = S(n+2)S(n). S(n) has length A000930(n).
Individual terms of a(n) may also be found by greedily writing n as a sum of entries of A000930. a(n) is 2 if the smallest term is 1, 3 if the smallest term is 2, and 1 otherwise.
(End)
a(n) = A005374(n+1) - A005374(n) - 2*(A202340(n+1) - 2). - Alan Michael Gómez Calderón, Jul 19 2025

Edited by N. J. A. Sloane, Oct 10 2007 and Aug 03 2014

A245554 A Rauzy fractal sequence: trajectory of 1 under morphism 1 -> 1,2,1,3; 2 -> 3; 3 -> 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Aug 03 2014

nonn

P. Arnoux and E. Harriss, What is a Rauzy Fractal?, Notices Amer. Math. Soc., 61 (No. 7, 2014), 768-770, also p. 704 and front cover.
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.
Index entries for sequences that are fixed points of mappings

Cf. A092782, A245553, A105083.

Nest[ Function[ l, {Flatten[(l /. {1 -> {1, 2, 1, 3}, 2 -> {3}, 3 -> {1}})] }], {1}, 9]

A245555 Trajectory of 1 under the morphism 1 -> 12, 2 -> 23, 3 -> 31.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Aug 03 2014

nonn

The morphism 1->12, 2->21 gives the {1,2} version of the Thue-Morse sequence A001285, cf. A010060.

The morphism 0->01, 1->12, 2->20 gives the generalized Thue-Morse sequence A071858.

Cf. A001285, A010060, A092782, A105083, A245553, A245554.

Essentially the same as A071858.

Nest[ Function[ l, {Flatten[(l /. {1 -> {1, 2}, 2 -> {2, 3}, 3 -> {3, 1}})] }], {1}, 9]

{a(n) = my(v = [1]); if( n<1, 0, while( #vMichael Somos, Aug 05 2014 */

a(n) = A071858(n+1) + 1. - Michel Dekking, Sep 29 2020

cp's OEIS Frontend

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.

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A073058 Define s(1)={1,2}, s(2)={1,3} and s(3)={1}. For a finite sequence A={a_1, ..., a_n}, with elements in {1,2,3}, define t(A) to be the concatenation of A, s(a_1), s(a_2), ... and s(a_n). Start with the sequence {1,2,3} and repeatedly apply t; limiting sequence is shown.

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A105083 Trajectory of 1 under the morphism 1 -> 12, 2 -> 3, 3 -> 1.

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A245554 A Rauzy fractal sequence: trajectory of 1 under morphism 1 -> 1,2,1,3; 2 -> 3; 3 -> 1.

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A245555 Trajectory of 1 under the morphism 1 -> 12, 2 -> 23, 3 -> 31.

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