A073058 -id:A073058 - cp's OEIS frontend

A103682 Trajectory of 1 under repeated applications of the morphism 1-> {1,2}, 2->{1,2,3}, 3->{1,1,3}.

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

Offset: 1

Roger L. Bagula, Mar 26 2005

nonn

Index entries for sequences that are fixed points of mappings

Cf. A073058.

s[1] = {1, 2}; s[2] = {1, 2, 3}; s[3] = {1, 1, 3}; t[a_] := Join[a, Flatten[s /@ a]]; a = t[t[t[t[t[{1}]]]]]

A105056 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->4, 4->{4,4,7,5}, 5->6, 6->7, 7->8, 8->{8,8,3,1}. 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, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 4, 4, 7, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 4, 4, 7, 5, 2, 3, 3, 4, 3, 4, 4, 4, 4, 7, 5, 3, 4, 4, 4, 4, 7, 5, 4, 4, 4, 7, 5, 4, 4, 7, 5, 4, 4, 7, 5, 4, 4, 7, 5, 8, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4

Offset: 0

Roger L. Bagula, Apr 04 2005

This sequence is the next level of substitution suggested in section 6 of the Kenyon paper. A tile exists at this level as well.

Richard Kenyon, The Construction of Self-Similar Tilings

Cf. A000120, A073058.

s[n_] := n /. {1 -> 2, 2 -> 3, 3 -> 4, 4 -> {4, 4, 7, 5}, 5 -> 6, 6 -> 7, 7 -> 8, 8 -> {8, 8, 3, 1}}; t[a_] := Join[a, Flatten[s /@ a]]; Flatten[ NestList[t, {1}, 6]]

A105061 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->4, 4->5, 5->{5,5,9,6}, 6->7, 7->8, 8->9, 9->10, 10->{10,10,4,1}. 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, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 5, 5, 9, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 5, 5, 9, 6, 2, 3, 3, 4

Offset: 1

Roger L. Bagula, Apr 05 2005

Level five bi-Kenyon substitution sequence.

Richard Kenyon, The Construction of Self-Similar Tilings

Cf. A000120, A073058.

s[n_] := n /. {1 -> 2, 2 -> 3, 3 -> 4, 4 -> 5, 5 -> {5, 5, 9, 6}, 6 -> 7, 7 -> 8, 8 -> 9, 9 -> 10, 10 -> {10, 10, 4, 1}}; t[a_] := Join[a, Flatten[s /@ a]]; Flatten[ NestList[t, {1}, 6]]

A105256 Sign doubling substitution of the Rauzy: 1->{1,2},2->{1,3},3->1 using a digraphy symmetry to the bi-Kenyon version (not a triangular nest of nests, but a straight level 5).

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Apr 14 2005

The French/ Siegel Rauzy substitution is: 1->{1,2} 2->{1,3} 3->{1} This is digraph symmetrical to the Kenyon type substitution : 1->{2} 2->{3} 3->{3,2,1} Looking at the digraph of: 1->{2} 2->{3} 3->{6,2,1} 4->{5} 5->{6} 6->{3,5,4} I get the same linked two triangle structure for this six-symbol substitution. The problem with the digraph approach is that the order is not specific as it is in actual substitutions.

"The Construction of Self-Similar Tilings", Richard Kenyon, Section 6

Cf. A073058, A105111.

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

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

A105265 Concatenation of letters of words obtained from axiom "1" and the iterates of the substitutions '1' -> "12", '2' -> "3", '3' -> "4", '4' -> "1".

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Apr 15 2005

Let W() be the substitution defined above. If we define the sequence S(n) by S(0) = {1}, S(n+1) = S(n) + W(S(n)), then this sequence is the limiting sequence of S(n) as n approaches infinity. - Charlie Neder, Jul 11 2018

Charlie Neder, Table of n, a(n) for n = 0..9999

Cf. A073058.

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

New name from Joerg Arndt, Jul 14 2018

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.

A104231 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->{3,3,5,4}, 4->5, 5->6, 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, 3, 5, 4, 1, 2, 2, 3, 2, 3, 3, 3, 3, 5, 4, 2, 3, 3, 3, 3, 5, 4, 3, 3, 3, 5, 4, 3, 3, 5, 4, 3, 3, 5, 4, 3, 3, 5, 4, 6, 5, 1, 2, 2, 3, 2, 3, 3, 3, 3, 5, 4, 2, 3, 3, 3, 3, 5, 4, 3, 3, 3, 5, 4, 3, 3, 5, 4, 3, 3, 5, 4, 3, 3, 5, 4, 6, 5, 2, 3, 3, 3, 3, 5, 4, 3, 3, 3, 5, 4, 3

Offset: 0

Roger L. Bagula, Apr 02 2005

This substitution was suggested by looking at output of the symbols of an actual Kenyon border tiling program.

Richard Kenyon, The Construction of Self-Similar Tilings

Cf. A073058, A103748.

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

A104232 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->{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, 2, 1, 1, 2, 2, 3, 2, 3, 3, 2, 1, 2, 3, 3, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 2, 3, 2, 3, 3, 2, 1, 2, 3, 3, 2, 1, 3, 2, 1, 2, 1, 3, 2, 2, 3, 3, 2, 1, 3, 2, 1, 2, 1, 3, 2, 3, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 3, 2, 3, 3, 2, 1, 2, 3, 3, 2, 1, 3, 2, 1, 2, 1

Offset: 0

Roger L. Bagula, Apr 02 2005

This substitution was suggested by looking at the Kenyon border tiling substitutions.

Richard Kenyon, The Construction of Self-Similar Tilings

Cf. A073058.

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

cp's OEIS Frontend

A103682 Trajectory of 1 under repeated applications of the morphism 1-> {1,2}, 2->{1,2,3}, 3->{1,1,3}.

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A105056 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->4, 4->{4,4,7,5}, 5->6, 6->7, 7->8, 8->{8,8,3,1}. 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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A105061 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->4, 4->5, 5->{5,5,9,6}, 6->7, 7->8, 8->9, 9->10, 10->{10,10,4,1}. 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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A105256 Sign doubling substitution of the Rauzy: 1->{1,2},2->{1,3},3->1 using a digraphy symmetry to the bi-Kenyon version (not a triangular nest of nests, but a straight level 5).

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A105265 Concatenation of letters of words obtained from axiom "1" and the iterates of the substitutions '1' -> "12", '2' -> "3", '3' -> "4", '4' -> "1".

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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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A104231 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->{3,3,5,4}, 4->5, 5->6, 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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A104232 Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->{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),...