A106749 -id:A106749 - cp's OEIS frontend

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

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

A106748 Define the morphism f: 1->113, 2->13, 3->2 and let a(0) = 1; then a(n+1) = f(a(n)).

Original entry on oeis.org

1, 113, 1131132, 1131132113113213, 113113211311321311311321131132131132, 113113211311321311311321131132131132113113211311321311311321131132131132113113213

Offset: 0

N. J. A. Sloane, May 16 2005

nonn

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.

Cf. A106749.

FromDigits /@ NestList[ Flatten[ # /. {1 -> {1, 1, 3}, 2 -> {1, 3}, 3 -> 2}] &, {1}, 5] (* Robert G. Wilson v, May 17 2005 *)

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

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

A106795 Fixed point of the morphism 1 -> 1,1,1,1,1,1,2,2,2,3; 2 -> 2,2,3,1,1,1,1; 3 -> 3,1,1,1,2,2, starting with a(0) = 1.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, May 17 2005

nonn

3-symbol substitution for the characteristic polynomial: x^3 + 9*x^2 - 3*x - 1.

			The first few steps of the substitution are:
Start: 1
Maps:
  1 --> 1 1 1 1 1 1 2 2 2 3
  2 --> 2 2 3 1 1 1 1
  3 --> 3 1 1 1 2 2
-------------
0:   (#=1)
  1
1:   (#=10)
  1111112223

G. C. Greubel, Table of n, a(n) for n = 0..10000
Victor F. Sirvent and Boris Solomyak, Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type. Canadian Mathematical Bulletin, 45(4), 2002, 697-710; (page 708 example 3). Also at ResearchGate
Index entries for sequences that are fixed points of mappings

Cf. A106749, A106796, A106797, A106798.

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

Edited by G. C. Greubel, Apr 03 2022

A106796 Fixed point of the morphism 1 -> 1,1,2; 2 -> 3; 3 -> 1,4; 4 -> 1, starting with a(0) = 1.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 1, 2, 1, 1, 2, 3

Offset: 0

Roger L. Bagula, May 17 2005

nonn

4-symbol substitution for the Pisot characteristic polynomial: x^4 - 2*x^2 - x - 1.

			The first few steps of the substitution are:
Start: 1
Maps:
  1 --> 1 1 2
  2 --> 3
  3 --> 1 4
  4 --> 1
-------------
0:   (#=1)
  1
1:   (#=3)
  112
2:   (#=7)
  1121123
3:   (#=16)
  1121123112112314
4:   (#=36)
  112112311211231411211231121123141121
5:   (#=82)
  1121123112112314112112311211231411211121123112112314112112311211231411211121123112

G. C. Greubel, Table of n, a(n) for n = 0..10000
Victor F. Sirvent and Boris Solomyak, Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type. Canadian Mathematical Bulletin, 45(4), 2002, 697-710; (page 709 example 3). Also at ResearchGate
Index entries for sequences that are fixed points of mappings

Cf. A106749, A106795, A106797, A106798.

s[1]= {1, 1, 2}; s[2]= {3}; s[3]= {1, 4}; s[4]= {1}; t[b_]:= Flatten[s /@ b];
a[0]= {1}; a[1]= t[p[0]]; a[n_]:= t[a[n-1]];
a[10]

Edited by G. C. Greubel, Apr 03 2022

A106797 Fixed point of the morphism 1 -> 1,1,1,1,2,2,3; 2 -> 4,1; 3 -> 2,1,1,1; 4 -> 1,2,1 starting with a(0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 3, 4, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 3, 4, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 3, 4, 1, 4, 1, 2

Offset: 0

Roger L. Bagula, May 17 2005

nonn

4-symbol substitution of the Pisot characteristic polynomial: x^4 - 4*x^3 - 6*x^2 - x - 1.

			The first few steps of the substitution are:
Start: 1
Maps:
  1 --> 1 1 1 1 2 2 3
  2 --> 4 1
  3 --> 2 1 1 1
  4 --> 1 2 1
-------------
0:   (#=1)
  1
1:   (#=7)
  1111223
2:   (#=36)
  111122311112231111223111122341412111

G. C. Greubel, Table of n, a(n) for n = 0..10000
Victor F. Sirvent and Boris Solomyak, Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type. Canadian Mathematical Bulletin, 45(4), 2002, 697-710; (page 709 example 4). Also at ResearchGate
Index entries for sequences that are fixed points of mappings

Cf. A106749, A106795, A106796, A106798.

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

Edited by G. C. Greubel, Apr 03 2022

A106798 Fixed point of the morphism 1 -> 3; 2 -> 1,2,2; 3 -> 1,2, starting with a(0) = 1.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, May 17 2005

3-symbol substitution for the characteristic polynomial: x^3 - 2*x^2 - x + 1.

			The first few steps of the substitution are:
Start: 1
Maps:
  1 --> 3
  2 --> 1 2 2
  3 --> 1 2
-------------
a(n) = p(2*n)
-------------
0:   (#=1) (p(0))
  1
1:   (#=2) (p(2))
  12
2:   (#=9) (p(4))
  123122122
3:   (#=45) (p(6))
  123122122312212312212231221221231221223122122

G. C. Greubel, Table of n, a(n) for n = 0..10000
Victor F. Sirvent and Boris Solomyak, Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type. Canadian Mathematical Bulletin, 45(4), 2002, 697-710. Also at ResearchGate
Index entries for sequences that are fixed points of mappings

Cf. A106749, A106795, A106796, A106797.

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

a(n) = p(2*n), where p(n) maps the fixed point morphism 1 -> 3; 2 -> 1,2,2; 3 -> 1,2, starting with p(0) = 1.

Edited by G. C. Greubel, Apr 03 2022

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

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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A106748 Define the morphism f: 1->113, 2->13, 3->2 and let a(0) = 1; then a(n+1) = f(a(n)).

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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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A106795 Fixed point of the morphism 1 -> 1,1,1,1,1,1,2,2,2,3; 2 -> 2,2,3,1,1,1,1; 3 -> 3,1,1,1,2,2, starting with a(0) = 1.

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A106796 Fixed point of the morphism 1 -> 1,1,2; 2 -> 3; 3 -> 1,4; 4 -> 1, starting with a(0) = 1.

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A106797 Fixed point of the morphism 1 -> 1,1,1,1,2,2,3; 2 -> 4,1; 3 -> 2,1,1,1; 4 -> 1,2,1 starting with a(0) = 1.

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A106798 Fixed point of the morphism 1 -> 3; 2 -> 1,2,2; 3 -> 1,2, starting with a(0) = 1.

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