A106748 -id:A106748 - cp's OEIS frontend

A106749 Define the morphism f: 1->113, 2->13, 3->2; sequence gives trajectory of 1 under f.

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

Offset: 0

N. J. A. Sloane, May 16 2005

E. Bombieri and J. Taylor, Which distribution of matter diffracts? An initial investigation, in International Workshop on Aperiodic Crystals (Les Houches, 1986), J. de Physique, Colloq. C3, 47 (1986), C3-19 to C3-28.

Index entries for sequences that are fixed points of mappings

Cf. A106748.

Nest[ Flatten[ # /. {1 -> {1, 1, 3}, 2 -> {1, 3}, 3 -> 2}] &, {1}, 6] (* Robert G. Wilson v, May 17 2005 *)

More terms from Robert G. Wilson v, May 17 2005

A103269 Apply the tribonacci morphism 1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1} n times to 1, and concatenate the resulting string.

Original entry on oeis.org

1, 12, 1213, 1213121, 1213121121312, 121312112131212131211213, 12131211213121213121121312131211213121213121, 121312112131212131211213121312112131212131211213121121312121312112131213121121312

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, May 16 2005

nonn

The number of letters in the n-th iteration is tribonacci(n+3) (that is, A000073(n+3)).

N. J. A. Sloane, Table of n, a(n) for n = 0..10 (shortened by _N. J. A. Sloane_, Jan 18 2019)
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320. See T on page 315.
Julien Cassaigne, Sebastien Ferenczi, and Luca Q. Zamboni, Imbalances in Arnoux-Rauzy sequences, Annales de l'institut Fourier, 50 (2000), 1265-1276.
David Damanik and Luca Q. Zamboni, Arnoux-Rauzy Subshifts: Linear Recurrence, Powers and Palindromes, arXiv:math/0208137 [math.CO], 2002.
C. Holton and L. Q. Zamboni, Directed Graphs and Substitutions, Theory Comput. Systems, 34 (2001) 545-564
S. Ito and M. Kimura, On Rauzy fractal, Japan J. Indust. Appl. Math. 8 (1991) 461-486.
A. Messaoudi, Propriétes arithmétiques et dynamiques du fractal de Rauzy, J. Th. Nombres Bordeaux, 10 (1998) 195-224.
Gérard Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France, 110 (1982), 147-178.
Bo Tan and Zhi-Ying Wen, Some properties of the Tribonacci sequence, European Journal of Combinatorics, 28 (2007) 1703-1719. See A_n.
Index entries for sequences that are fixed points of mappings

A092782 is limit of these strings.

Cf. A000073, A106748, A106749, A003842, A106750, A092782, A317196.

FromDigits /@ NestList[ Flatten[ # /. {1 -> {1, 2}, 2 -> {1, 3}, 3 -> 1}] &, {1}, 7]
(* Second program: *)
FromDigits /@ SubstitutionSystem[{1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1}}, {1}, 7] (* Jean-François Alcover, Nov 12 2018 *)

Definition edited by N. J. A. Sloane, Aug 06 2018

A106750 Define the "Fibonacci" morphism f: 1->12, 2->1 and let a(0) = 2; then a(n+1) = f(a(n)).

Original entry on oeis.org

2, 1, 12, 121, 12112, 12112121, 1211212112112, 121121211211212112121, 1211212112112121121211211212112112, 1211212112112121121211211212112112121121211211212112121

Offset: 0

N. J. A. Sloane, May 16 2005. Initial term 2 added by N. J. A. Sloane, Jul 05 2012

nonn

a(n) converges to the Fibonacci word A003842.

a(n) has length Fibonacci(n+1) (cf. A000045).

Berstel, Jean. "Fibonacci words—a survey." In The book of L, pp. 13-27. Springer Berlin Heidelberg, 1986.
E. Bombieri and J. Taylor, Which distribution of matter diffracts? An initial investigation, in International Workshop on Aperiodic Crystals (Les Houches, 1986), J. de Physique, Colloq. C3, 47 (1986), C3-19 to C3-28.

N. J. A. Sloane, Table of n, a(n) for n = 0..15

Cf. A106748, A106749, A003842, A000045, A213975, A213976.

FromDigits /@ NestList[ Flatten[ # /. {1 -> {1, 2}, 2 -> 1}] &, {2}, 8] (* Robert G. Wilson v, May 17 2005 *)

More terms from Robert G. Wilson v, May 17 2005

A107292 3-symbol substitution with characteristic real root polynomial:m x^3-2x^2-2x+2.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, May 20 2005

This is a real root cubic:{{x -> -1.17009}, {x -> 0.688892}, {x -> 2.48119}} like the Bombieri aperiodic: a Bombieri silver Isomer substitution: ( same characteristic polynomial) 1->{3},2->{2,1,2},3->{1,2,2,1}

Cf. A106748, A106749.

s[1] = {1, 3, 3, 1}; s[2] = {3, 1, 3}; s[3] = {2}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]] aa = p[5]

1->{1, 3, 3, 1}, 2->{3, 1, 3}, 3->{2}

A107296 Three-symbol substitution with real Pisot characteristic polynomial: x^3-3*x^2-x-2.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, May 20 2005

Bombieri type Real Roots: {{x -> -0.860806}, {x -> 0.745898}, {x -> 3.11491}} Matrix isomer: 1->{3},{2->{2,1,2,2},3->{1,2} I found this while trying to get a substitution for the Frougny real root characteristic polynomial: x^3-3*x^2+1

Cf. A106748, A106749.

s[1] = {1, 3, 1, 1}; s[2] = {1, 3}; s[3] = {2}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]] aa = p[4]

1->{1, 3, 1, 1}, 2->{1, 3}, 2->{2}

A107297 Limit of the string substitution 1->{1, 3, 1, 1}, 2->{1}, 3->{2}.

Original entry on oeis.org

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

Offset: 1

Roger L. Bagula, May 20 2005

nonn

Cf. A106748, A106749.

s[1] = {1, 3, 1, 1}; s[2] = {1}; s[3] = {2};
t[a_] := Flatten[s /@ a];
p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]]
aa = p[5]

Edited by and meaningful name from Joerg Arndt, Jun 30 2023

cp's OEIS Frontend

A106749 Define the morphism f: 1->113, 2->13, 3->2; sequence gives trajectory of 1 under f.

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A103269 Apply the tribonacci morphism 1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1} n times to 1, and concatenate the resulting string.

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A106750 Define the "Fibonacci" morphism f: 1->12, 2->1 and let a(0) = 2; then a(n+1) = f(a(n)).

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A107292 3-symbol substitution with characteristic real root polynomial:m x^3-2*x^2-2*x+2.

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Programs

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A107296 Three-symbol substitution with real Pisot characteristic polynomial: x^3-3*x^2-x-2.

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Programs

Mathematica

Formula

A107297 Limit of the string substitution 1->{1, 3, 1, 1}, 2->{1}, 3->{2}.

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Programs

Mathematica

Extensions

A107292 3-symbol substitution with characteristic real root polynomial:m x^3-2x^2-2x+2.