A103269 -id:A103269 - 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

A113535 Ascending descending base exponent transform of the tribonacci substitution (A100619).

Original entry on oeis.org

1, 3, 8, 19, 32, 9, 11, 16, 26, 19, 29, 24, 47, 70, 28, 31, 58, 89, 35, 50, 65, 108, 65, 51, 52, 90, 101, 82, 101, 88, 122, 63, 81, 92, 153, 110, 89, 125, 110, 92, 101, 155, 90, 127, 196, 142, 87, 138, 207, 112, 112, 135, 217, 150, 124, 115, 204, 245, 139, 158, 189, 268, 121, 155, 154

Offset: 1

Jonathan Vos Post, Jan 13 2006

Sirvent comments that in spite of the similarity of this map to the one in A092782, the two sequences have very different properties. They have different complexities, different Rauzy fractals, etc.

			a(1) = A100619(1)^A100619(1) = 1^1 = 1.
a(2) = A100619(1)^A100619(2) + A100619(2)^A100619(1) = 1^2 + 2^1 = 3.
a(3) = 1^3 + 2^2 + 3^1 = 8.
a(4) = 1^1 + 2^3 + 3^2 + 1^1 = 19.
a(5) = 1^1 + 2^1 + 3^3 + 1^2 + 1^1 = 32.
a(6) = 1^1 + 2^1 + 3^1 + 1^3 + 1^2 + 1^1 = 9.
a(7) = 1^2 + 2^1 + 3^1 + 1^1 + 1^3 + 1^2 + 2^1 = 11.
a(8) = 1^1 + 2^2 + 3^1 + 1^1 + 1^1 + 1^3 + 2^2 + 1^1 = 16.
a(9) = 1^1 + 2^1 + 3^2 + 1^1 + 1^1 + 1^1 + 2^3 + 1^2 + 2^1 = 26.
a(10) = 1^1 + 2^2 + 3^1 + 1^2 + 1^1 + 1^1 + 2^1 + 1^3 + 2^2 + 1^1 = 19.
a(11) = 1^2 + 2^1 + 3^2 + 1^1 + 1^2 + 1^1 + 2^1 + 1^1 + 2^3 + 1^2 + 2^1 = 29.
a(12) = 1^3 + 2^2 + 3^1 + 1^2 + 1^1 + 1^2 + 2^1 + 1^1 + 2^1+ 1^3 + 2^2 + 3^1 = 24.

G. C. Greubel, Table of n, a(n) for n = 1..500
V. F. Sirvent, Semigroups and the self-similar structure of the flipped tribonacci substitution, Applied Math. Letters, 12 (1999), 25-29. [Contains many further references.]

Cf. A100619, A092782, A103269, A113320, A005408, A113122, A113153, A113154, A113336, A113271, A113258, A113257, A113231, A087316, A113208, A113498.

A100619:= Nest[Function[l, {Flatten[(l /. {1 -> {1, 2}, 2 -> {3, 1}, 3 -> {1}})]}], {1}, 8][[1]]; Table[Sum[(A100619[[k]])^(A100619[[n-k+1]]), {k, 1, n}], {n, 1, 100}] (* G. C. Greubel, May 18 2017 *)

a(n) = Sum_{k=1..n} A100619(k)^(A100619(n-k-1)). - G. C. Greubel, May 18 2017

Terms a(13) to a(50) from G. C. Greubel, May 18 2017

Terms a(51) onward added by G. C. Greubel, Jan 03 2019

A317953 Apply the morphism 1 -> {1, 2}, 2 -> {3,1}, 3 -> {1} n times to 1, and concatenate the resulting string.

Original entry on oeis.org

1, 12, 1231, 1231112, 1231112121231, 123111212123112311231112, 12311121212311231123111212311121231112121231, 123111212123112311231112123111212311121212311231112121231123111212123112311231112

Offset: 1

N. J. A. Sloane, Aug 21 2018

nonn

For the tribonacci word A092782, each block b(n) (see A103269) is the concatenation of the three previous blocks: b(n) = b(n-1).b(n-2).b(n-3). Instead, here we have a(n) = a(n-1).a(n-3).a(n-2), as can be seen in the examples section below.

			a(1): 1,
a(2): 12,
a(3): 1231,
a(4): 1231112,
a(5): 1231112121231,
a(6): 123111212123112311231112,
a(7): 12311121212311231123111212311121231112121231,
equals a(6).a(4).a(5), look:
a(6):123111212123112311231112,
a(4):                        1231112,
a(5):                               1231112121231,
a(8): 123111212123112311231112123111212311121212311231112121231123111212123112311231112
equals a(7).a(5).a(6), look:
a(7): 12311121212311231123111212311121231112121231,
a(5):                                             1231112121231,
a(6):                                                          123111212123112311231112,

V. F. Sirvent, Semigroups and the self-similar structure of the flipped tribonacci substitution, Applied Math. Letters, 12 (1999), 25-29.

Cf. A100619 (the limiting string), A277735, A317953.

A103269 is the analog for the word A092782.

cp's OEIS Frontend

A317197 a(n) is the concatenation of A103269(n-i) for i = 0,1,2,...,n-1.

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A317199 Bo Tan et al.'s string E_n, defined by A_n = A103269(n) = D_{n-1}E_n = A317197(n-1)E_n for n >= 2, with E_1 = 12.

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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.

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A113535 Ascending descending base exponent transform of the tribonacci substitution (A100619).

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A317953 Apply the morphism 1 -> {1, 2}, 2 -> {3,1}, 3 -> {1} n times to 1, and concatenate the resulting string.

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A317196 a(n) = concatenation of n-th stage of the trajectory of 2 under the tribonacci morphism 1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1}.

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