A103684 -id:A103684 - cp's OEIS frontend

A138056 Levels of substitution A103684 (based on the morphism f: 1->{1,2}, 2->{1,3}, 3->{3}) like Markov substitution taken as polynomials p(x,n)]and coefficients of the differential polynomials returned as q(x,n) =dp(x,n)dx coefficients (first zero omitted).

2, 2, 2, 9, 2, 2, 9, 4, 10, 6, 2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24, 2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24, 13, 28, 15, 48, 17, 36, 19, 20, 42, 22, 69, 2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24, 13, 28, 15, 48, 17, 36, 19, 20, 42, 22, 69, 24, 50, 26, 81, 28, 58, 30, 31, 64, 33, 102, 35

Offset: 1

Roger L. Bagula, May 02 2008

			{2},
{2, 2, 9},
{2, 2, 9, 4, 10, 6},
{2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24},
{2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24, 13, 28, 15, 48, 17, 36, 19, 20, 42, 22, 69},
{2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24, 13, 28, 15, 48, 17, 36, 19, 20, 42, 22, 69, 24, 50, 26, 81, 28, 58, 30, 31, 64, 33, 102, 35, 72, 37, 76, 39, 120, 41, 84, 43}

Cf. A103684.

s[1] = {1, 2}; s[2] = {1, 3}; s[3] = {1};
t[a_] := Flatten[s /@ a];
p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]];
(*A103684*);
a = Table[p[n], {n, 0, 10}];
Flatten[a];
b = Table[CoefficientList[D[Apply[Plus, Table[a[[n]][[m]]*x^( m - 1), {m, 1, Length[a[[n]]]}]], x], x], {n, 1, 11}];
Flatten[b]
Table[Apply[Plus, CoefficientList[D[Apply[Plus, Table[a[[n]][[m]]* x^(m - 1), {m, 1, Length[a[[n]]]}]], x], x]], {n, 1, 11}];

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 *)

A103685 Consider the morphism 1->{1,2}, 2->{1,3}, 3->{1}; a(n) is the total number of '3' after n substitutions.

Original entry on oeis.org

0, 0, 1, 5, 17, 51, 147, 419, 1191, 3383, 9607, 27279, 77455, 219919, 624415, 1772895, 5033759, 14292287, 40579903, 115217983, 327136895, 928835455, 2637230207, 7487852799, 21260161279, 60363694335, 171389837823, 486624896511, 1381667623423, 3922950583295

Offset: 0

Roger L. Bagula, Mar 26 2005

Examples of the morphism starting with {1} are shown in A103684. Counting the total number of '1' in rows 1 to n of A103684 yields 1, 3, 8,... = A073357(n+1),
counting the total number of '2' in rows 1 to n yields 0, 1, 4,.. = A115390(n+1),
and counting the total number '3' in rows 1 to n yields a(n), the sequence here.
Inverse binomial transform yields 0, 0, 1, 2, 3, 6, 11, 20,..., a variant of A001590 [Nov 18 2010]

Index entries for linear recurrences with constant coefficients, signature (5,-8,6,-2).

Cf. A073058, A103684.

LinearRecurrence[{5,-8,6,-2},{0,0,1,5},30] (* Harvey P. Dale, Nov 10 2011 *)

a(n)= +5*a(n-1) -8*a(n-2) +6*a(n-3) -2*a(n-4) = a(n-1)+A115390(n). [Nov 18 2010]

G.f.: x^2 / ( (x-1)*(2*x^3-4*x^2+4*x-1) ). [Nov 18 2010]

Depleted by the information already in A073357 and A115390; corrected image of {2} in the defn. - The Assoc. Eds. of the OEIS, Nov 18 2010

A105111 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->{3,5,4}, 4->5, 5->6, 6->{6,2,1}. First row is 1. If current row is a,b,c,..., then the next row is a,b,c,...,f(a),f(b),f(c),...

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 3, 5, 4, 1, 2, 2, 3, 2, 3, 3, 3, 5, 4, 2, 3, 3, 3, 5, 4, 3, 3, 5, 4, 3, 5, 4, 3, 5, 4, 6, 5, 1, 2, 2, 3, 2, 3, 3, 3, 5, 4, 2, 3, 3, 3, 5, 4, 3, 3, 5, 4, 3, 5, 4, 3, 5, 4, 6, 5, 2, 3, 3, 3, 5, 4, 3, 3, 5, 4, 3, 5, 4, 3, 5, 4, 6, 5, 3, 3, 5, 4, 3, 5, 4, 3, 5, 4, 6, 5, 3, 5

Offset: 0

Roger L. Bagula, Apr 07 2005

6-symbol substitution based on the second type Rauzy substitution that gives a tile in the Kenyon boundary method.

Richard Kenyon, The Construction of Self-Similar Tilings

Cf. A073058, A103684.

s[n_] := n /. {1 -> 2, 2 -> 3, 3 -> {3, 5, 4}, 4 -> 5, 5 -> 6, 6 -> {6, 2, 1}}; t[a_] := Join[a, Flatten[s /@ a]]; Flatten[ NestList[t, {1}, 5]]

A138060 Triangle read by rows: row 1 = {1}; for n>1, row n is obtained from row n-1 by applying the morphism 1->1,2; 2->3; 3->4; 4->1.

Original entry on oeis.org

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

Offset: 1

Roger L. Bagula, May 02 2008

			{1},
{1, 2},
{1, 2, 3},
{1, 2, 3, 4},
{1, 2, 3, 4, 1},
{1, 2, 3, 4, 1, 1, 2},
{1, 2, 3, 4, 1, 1, 2, 1, 2, 3},
{1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4},
{1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1},
{1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 1, 2, 3, 4, 1, 1, 2},
{1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3}

Row sums are A138289. Rows converge to A138297.

Cf. A103684.

SubstitutionSystem[{1 -> {1, 2}, 2 -> {3}, 3 -> {4}, 4 -> {1}}, {1}, 10] // Flatten (* Jean-François Alcover, Jul 01 2023 *)

Edited by N. J. A. Sloane, May 06 2008

A105112 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->{3,5,5,4}, 4->5, 5->6, 6->{6,2,2,1}. First row is 1. If current row is a,b,c,..., then the next row is a,b,c,...,f(a),f(b),f(c),...

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 3, 5, 5, 4, 1, 2, 2, 3, 2, 3, 3, 3, 5, 5, 4, 2, 3, 3, 3, 5, 5, 4, 3, 3, 5, 5, 4, 3, 5, 5, 4, 3, 5, 5, 4, 6, 6, 5, 1, 2, 2, 3, 2, 3, 3, 3, 5, 5, 4, 2, 3, 3, 3, 5, 5, 4, 3, 3, 5, 5, 4, 3, 5, 5, 4, 3, 5, 5, 4, 6, 6, 5, 2, 3, 3, 3, 5, 5, 4, 3, 3, 5, 5, 4, 3, 5, 5, 4, 3, 5, 5

Offset: 0

Roger L. Bagula, Apr 07 2005

Improved version of bi-Kenyon 6-symbol substitution.

Richard Kenyon, The Construction of Self-Similar Tilings

Cf. A103684.

s[n_] := n /. {1 -> 2, 2 -> 3, 3 -> {3, 5, 5, 4}, 4 -> 5, 5 -> 6, 6 -> {6, 2, 2, 1}}; t[a_] := Join[a, Flatten[s /@ a]]; Flatten[ NestList[t, {1}, 5]]

A248335 A recursive sequence generated by an L-system defined in comments.

Original entry on oeis.org

1, 23, 3445, 45565667, 5667677867787889, 6778788978898990788989908990901, 7889899089909018990901901112899090190111290111211223, 89909019011129011121122390111211223112232323349011121122311223232334112232323342323343445

Offset: 1

Felix Fröhlich, Oct 26 2014

The L-system producing the sequence is defined as follows:
Alphabet: 1 2 3 4 5 6 7 8 9 0
Initiator: 1
Production rules: (1 --> 23), (2 --> 34), (3 --> 45), (4 --> 56), (5 --> 67), (6 --> 78), (7 --> 89), (8 --> 90), (9 --> 1), (0 --> 12).

Felix Fröhlich, C++ program for this sequence
Wikipedia, L-system

Cf. A057985, A073058, A103684.

A103956 A nonsense sequence.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Mar 30 2005

Cf. A073058, A103684.

Conway[1] = Conway[2] = 1;
Conway[n_Integer?Positive] := Conway[n] = Conway[Conway[n - 1]] + Conway[n - Conway[n - 1]]
s[1] = {1, 2}; s[2] = {1, 3}; s[3] = {1};
t[a_] := Join[a, Flatten[s /@ a]];
p[0] = {1}; p[1] = t[{1}];
p[n_] := t[p[n - 1]]
aa = Flatten[Table[p[If[n > 0, Conway[n], n]], {n, 0, 7}]]

1-> {1, 2} 2->{1, 3} 3->1 nested nest of substitution list are taken in a chaotic order.

A103957 A nonsense sequence.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Mar 30 2005

Cf. A073058, A103684.

Hofstadter[n_Integer? Positive] := Hofstadter[n] = Hofstadter[n - Hofstadter[n - 1]] + Hofstadter[n - Hofstadter[n - 2]];
Hofstadter[0] = Hofstadter[1] = 1;
s[1] = {1, 2}; s[2] = {1, 3}; s[3] = {1};
t[a_] := Join[a, Flatten[s /@ a]];
p[0] = {1}; p[1] = t[{1}];
p[n_] := t[p[n - 1]];
Flatten[Table[p[If[n > 0, Hofstadter[n], n]], {n, 0, 7}]]

Involves substitutions 1-> {1, 2}, 2->{1, 3}, 3->1.

A105103 A bi-Rauzy Pisot substitution.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 6, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 6, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 6, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 6, 1, 2, 1, 3, 1, 2, 6, 1, 2, 1, 3, 1

Offset: 0

Roger L. Bagula, Apr 07 2005

Cf. A073058, A103684.

s[1] = {1, 2}; s[2] = {1, 3}; s[3] = {6}; s[4] = {4, 5}; s[5] = {4, 6}; s[6] = {1}; t[a_] := Join[a, Flatten[s /@ a]]; p[0] = {1}; p[1] = t[{1}]; p[n_] := t[p[n - 1]] aa = Flatten[Table[p[n], {n, 0, 6}]]

1->{1, 2} 2->{1, 3} 3->6 4->{4, 5} 5->{4, 6) 6->1

cp's OEIS Frontend

A138056 Levels of substitution A103684 (based on the morphism f: 1->{1,2}, 2->{1,3}, 3->{3}) like Markov substitution taken as polynomials p(x,n)]and coefficients of the differential polynomials returned as q(x,n) =dp(x,n)dx coefficients (first zero omitted).

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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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A103685 Consider the morphism 1->{1,2}, 2->{1,3}, 3->{1}; a(n) is the total number of '3' after n substitutions.

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A105111 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->{3,5,4}, 4->5, 5->6, 6->{6,2,1}. First row is 1. If current row is a,b,c,..., then the next row is a,b,c,...,f(a),f(b),f(c),...

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A138060 Triangle read by rows: row 1 = {1}; for n>1, row n is obtained from row n-1 by applying the morphism 1->1,2; 2->3; 3->4; 4->1.

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A105112 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->{3,5,5,4}, 4->5, 5->6, 6->{6,2,2,1}. First row is 1. If current row is a,b,c,..., then the next row is a,b,c,...,f(a),f(b),f(c),...

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A248335 A recursive sequence generated by an L-system defined in comments.

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A103956 A nonsense sequence.

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A103957 A nonsense sequence.

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A105103 A bi-Rauzy Pisot substitution.