A317950 -id:A317950 - cp's OEIS frontend

A080843 Tribonacci word: limit S(infinity), where S(0) = 0, S(1) = 0,1, S(2) = 0,1,0,2 and for n >= 0, S(n+3) = S(n+2) S(n+1) S(n).

0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2

Offset: 0

N. J. A. Sloane, Mar 29 2003

An Arnoux-Rauzy or episturmian word.
From N. J. A. Sloane, Jul 10 2018: (Start)
The initial terms in a form suitable for copying:
0,1,0,2,0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,1,0,0,1,0,2,0,1,
0,1,0,2,0,1,0,0,1,0,2,0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,1,
0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,1,
0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,
2,0,1,0,2,0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,1,0,0,1,0,2,0,
1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,
1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,
1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,
...
Let TTW(a,b,c) denote this sequence written over the alphabet {a,b,c}. It begins:
a,b,a,c,a,b,a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,a,c,a,b,a,c,a,b,a,a,b,a,c,a,b,
a,b,a,c,a,b,a,a,b,a,c,a,b,a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,a,c,a,b,a,c,a,b,
a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,a,c,a,b,a,c,a,b,
a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,a,c,a,b,a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,a,
c,a,b,a,c,a,b,a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,a,c,a,b,a,c,a,b,a,a,b,a,c,a,
b,a,b,a,c,a,b,a,a,b,a,c,a,b,a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,a,c,a,b,a,c,a,
b,a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,a,c,a,b,a,c,a,
b,a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,a,c,a,b,a,a,b,a,c,a,b,a,b,a,c,a,b,a,a,b,
... (End)
From Wolfdieter Lang, Aug 14 2018: (Start)
The substitution sequence 0 -> 0, 1; 1-> 0, 2; 2 -> 0  read as an irregular triangle with rows l >= 1 and length T(l+2), with the tribonacci numbers T = A000073, leads to the tribonacci tree TriT with level TriT(l) for l >= 1 given by a(0), a(1), ..., a(T(l+2)-1).
  E.g., l = 4:  0 1  0 2  0 1  0 with T(6) = 7 leaves (nodes). See the example below.
This tree can be used to find the tribonacci representation of nonnegative n given in A278038, call it ZTri(n) (Z for generalized Zeckendorf), by replacing every 2 by 1, and reading from bottom to top, omitting the final 0, except for n = 0 which is represented by 0. See the example below. (End)

			From _Joerg Arndt_, Mar 12 2013: (Start)
The first few steps of the substitution are
Start: 0
Rules:
  0 --> 01
  1 --> 02
  2 --> 0
-------------
0:   (#=1)
  0
1:   (#=2)
  01
2:   (#=4)
  0102
3:   (#=7)
  0102010
4:   (#=13)
  0102010010201
5:   (#=24)
  010201001020101020100102
6:   (#=44)
  01020100102010102010010201020100102010102010
7:   (#=81)
  010201001020101020100102010201001020101020100102010010201010201001020102010010201
(End)
From _Wolfdieter Lang_, Aug 14 2018: (Start)
The levels l of the tree TriT begin (the branches (edges) have been omitted):
Substitution rule: 0 -> 0 1; 1 -> 0 2; 2 -> 0.
l=1:                                 0
l=2:                  0                                 1
l=3:             0             1                  0             2
l=4:         0      1       0     2          0       1          0
l=5:      0    1  0   2   0   1   0        0   1   0   2      0    1
...
----------------------------------------------------------------------------------
n =       0    1  2   3   4   5   6        7   8   9  10     11   12
The tribonacci representation of n >= 0 (A278038; here at level 5 for n = 0.. 12) is obtained by reading from bottom to top (along the branches not shown) replacing 2 with 1, omitting the last 0 except for n = 0.
          0    1  0   1   0   1   0        0   1   0  1      0    1
                  1   1   0   0   1        0   0   1  1      0    0
                          1   1   1        0   0   0  0      1    1
                                           1   1   1  1      1    1
E.g., ZTri(9) = A278038(9) = 1010. (End)

The entry A092782 has a more complete list of references and links. - N. J. A. Sloane, Aug 17 2018
J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 246.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
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.
D. Damanik and L. Q. Zamboni, Arnoux-Rauzy subshifts: linear recurrence, powers and palindromes, arXiv:math/0208137 [math.CO], 2002.
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]
Robbert Fokkink and Dan Rust, Queen reflections: a modification of Wythoff Nim, Int'l J. Game Theory (2022).
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv:1810.09787v1 [math.NT], 2018.
Joseph Meleshko, Pascal Ochem, Jeffrey Shallit, and Sonja Linghui Shan, Pseudoperiodic Words and a Question of Shevelev, arXiv:2207.10171 [math.CO], 2022.
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.
Bo Tan and Zhi-Ying Wen, Some properties of the Tribonacci sequence, European Journal of Combinatorics, 28 (2007) 1703-1719.
O. Turek, Abelian Complexity Function of the Tribonacci Word, J. Int. Seq. 18 (2015) # 15.3.4
Index entries for sequences that are fixed points of mappings

Cf. A003849 (the Fibonacci word), A092782.
See A092782 for a version over the alphabet {1,2,3}.
See A278045 for another construction.
Cf. A000073, A278038.
First differences: A317950. Partial sums: A319198.

M:=17; S[1]:=`0`; S[2]:=`01`; S[3]:=`0102`;
for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:
t0:=S[M]: l:=length(t0); for i from 1 to l do lprint(i,substring(t0,i..i)); od:
# N. J. A. Sloane, Nov 01 2006
# A version that uses the letters a,b,c:
M:=10; B[1]:=`a`; B[2]:=`ab`; B[3]:=`abac`;
for n from 4 to M do B[n]:=cat(B[n-1], B[n-2], B[n-3]); od:
B[10]; # N. J. A. Sloane, Oct 30 2018

Nest[Flatten[ # /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0}}] &, {0}, 8] (* updated by Robert G. Wilson v, Nov 07 2010 *)
SubstitutionSystem[{0->{0,1},1->{0,2},2->{0}},{0},{8}]//Flatten (* Harvey P. Dale, Nov 21 2021 *)

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=[0];  for (k=1, 10, t=strsub( t, [[0,1], [0,2], [0]], 0 ) );  t
\\ Joerg Arndt, Sep 14 2013

Fixed point of morphism 0 -> 0, 1; 1 -> 0, 2; 2 -> 0.

a(n) = A092782(n+1) - 1. - Joerg Arndt, Sep 14 2013

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A080843 Tribonacci word: limit S(infinity), where S(0) = 0, S(1) = 0,1, S(2) = 0,1,0,2 and for n >= 0, S(n+3) = S(n+2) S(n+1) S(n).

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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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions