cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A287174 2-limiting word of the morphism 0->10, 1->20, 2->0.

Original entry on oeis.org

2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0
Offset: 1

Views

Author

Clark Kimberling, May 22 2017

Keywords

Comments

Starting with 0, the first 5 iterations of the morphism yield words shown here:
1st: 10
2nd: 2010
3rd: 0102010
4th: 1020100102010
5th: 201001020101020100102010
The 2-limiting word is the limit of the words for which the number of iterations is congruent to 2 mod 3.
Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
U = 1.8392867552141611325518525646532866...,
V = U^2 = 3.3829757679062374941227085364...,
W = U^3 = 6.2222625231203986266745611011....
If n >=2, then u(n) - u(n-1) is in {1,2}, v(n) - v(n-1) is in {2,3,4}, and w(n) - w(n-1) is in {4,6,7}.

Examples

			2nd iterate: 2010
5th iterate: 201001020101020100102010

Links

Crossrefs

Cf. A286998, A287111, A287175, A287176, A287177.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 11] (* A287174 *)
Flatten[Position[s, 0]] (* A287175 *)
Flatten[Position[s, 1]] (* A287176 *)
Flatten[Position[s, 2]] (* A287177 *)

A287176 Positions of 1 in A287174.

Original entry on oeis.org

3, 6, 10, 12, 16, 19, 23, 26, 30, 32, 36, 39, 43, 47, 50, 54, 56, 60, 63, 67, 69, 73, 76, 80, 84, 87, 91, 93, 97, 100, 104, 107, 111, 113, 117, 120, 124, 128, 131, 135, 137, 141, 144, 148, 151, 155, 157, 161, 164, 168, 172, 175, 179, 181, 185, 188, 192, 194
Offset: 1

Views

Author

Clark Kimberling, May 22 2017

Keywords

Crossrefs

Cf. A287174, A287175, A287177.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 11] (* A287174 *)
Flatten[Position[s, 0]] (* A287175 *)
Flatten[Position[s, 1]] (* A287176 *)
Flatten[Position[s, 2]] (* A287177 *)

A287177 Positions of 2 in A287174.

Original entry on oeis.org

1, 8, 14, 21, 28, 34, 41, 45, 52, 58, 65, 71, 78, 82, 89, 95, 102, 109, 115, 122, 126, 133, 139, 146, 153, 159, 166, 170, 177, 183, 190, 196, 203, 207, 214, 220, 227, 234, 240, 247, 251, 258, 264, 271, 275, 282, 288, 295, 302, 308, 315, 319, 326, 332, 339
Offset: 1

Views

Author

Clark Kimberling, May 22 2017

Keywords

Links

Crossrefs

Cf. A287174, A287175, A287176.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 11] (* A287174 *)
Flatten[Position[s, 0]] (* A287175 *)
Flatten[Position[s, 1]] (* A287176 *)
Flatten[Position[s, 2]] (* A287177 *)

A287200 2-limiting word of the morphism 0->10, 1->22, 2->0, starting with 0.

Original entry on oeis.org

2, 2, 1, 0, 2, 2, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 0, 2, 2, 1, 0, 2, 2, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 1, 0, 2, 2, 1, 0, 2, 2, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 0
Offset: 1

Views

Author

Clark Kimberling, May 23 2017

Keywords

Comments

Starting with 0, the first 5 iterations of the morphism yield words shown here:
1st: 10
2nd: 2210
3rd: 002210
4th: 1010002210
5th: 221022101010002210
The 2-limiting word is the limit of the words for which the number of iterations is congruent to 2 mod 3.
Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
U = 2.28537528186132044169516884721360670506...,
V = 3.87512979416277882597397059430967806752...,
W = 3.28537528186132044169516884721360670506...
If n >=2, then u(n) - u(n-1) is in {1,2,4}, v(n) - v(n-1) is in {2,4,6}, and w(n) - w(n-1) is in {1,3,5,9}.

Examples

			2nd iterate: 2210
5th iterate: 221022101010002210

Links

Crossrefs

Cf. A287175, A287179, A287201, A287202.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 2}, 2 -> 0}] &, {0}, 11] (* A287200 *)
Flatten[Position[s, 0]] (* A287201 *)
Flatten[Position[s, 1]] (* A287202 *)
Flatten[Position[s, 2]] (* A287203 *)

Extensions

Definition corrected by Georg Fischer, May 27 2021

A287321 Positions of 0 in A287320.

Original entry on oeis.org

1, 2, 6, 7, 8, 12, 16, 20, 22, 24, 25, 26, 30, 31, 32, 36, 37, 38, 42, 46, 50, 52, 54, 55, 56, 60, 64, 68, 70, 72, 73, 74, 78, 82, 86, 88, 90, 91, 92, 96, 98, 100, 101, 102, 106, 108, 110, 111, 112, 116, 117, 118, 122, 123, 124, 128, 132, 136, 138, 140, 141
Offset: 1

Views

Author

Clark Kimberling, May 23 2017

Keywords

Links

Crossrefs

Cf. A287320, A287322, A287323.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 2}, 2 -> 0}] &, {0}, 9] (* A287175 *)
Flatten[Position[s, 0]] (* A287321 *)
Flatten[Position[s, 1]] (* A287322 *)
Flatten[Position[s, 2]] (* A287323 *)

A287322 Positions of 1 in A287320.

Original entry on oeis.org

5, 11, 15, 19, 21, 23, 29, 35, 41, 45, 49, 51, 53, 59, 63, 67, 69, 71, 77, 81, 85, 87, 89, 95, 97, 99, 105, 107, 109, 115, 121, 127, 131, 135, 137, 139, 145, 151, 157, 161, 165, 167, 169, 175, 181, 187, 191, 195, 197, 199, 205, 209, 213, 215, 217, 223, 227
Offset: 1

Views

Author

Clark Kimberling, May 23 2017

Keywords

Links

Crossrefs

Cf. A287320, A287321, A287323.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 2}, 2 -> 0}] &, {0}, 9] (* A287175 *)
Flatten[Position[s, 0]] (* A287321 *)
Flatten[Position[s, 1]] (* A287322 *)
Flatten[Position[s, 2]] (* A287323 *)

A287323 Positions of 2 in A287320.

Original entry on oeis.org

3, 4, 9, 10, 13, 14, 17, 18, 27, 28, 33, 34, 39, 40, 43, 44, 47, 48, 57, 58, 61, 62, 65, 66, 75, 76, 79, 80, 83, 84, 93, 94, 103, 104, 113, 114, 119, 120, 125, 126, 129, 130, 133, 134, 143, 144, 149, 150, 155, 156, 159, 160, 163, 164, 173, 174, 179, 180, 185
Offset: 1

Views

Author

Clark Kimberling, May 23 2017

Keywords

Links

Crossrefs

Cf. A287320, A287321, A287322.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 2}, 2 -> 0}] &, {0}, 9] (* A287175 *)
Flatten[Position[s, 0]] (* A287321 *)
Flatten[Position[s, 1]] (* A287322 *)
Flatten[Position[s, 2]] (* A287323 *)

A287331 1-limiting word of the morphism 0->10, 1->22, 2->0, starting with 0.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 2, 2, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 1, 0, 2, 2, 1, 0, 2, 2, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 1, 0, 2, 2, 1, 0
Offset: 1

Views

Author

Clark Kimberling, May 23 2017

Keywords

Comments

Starting with 0, the first 5 iterations of the morphism yield words shown here:
1st: 10
2nd: 2210
3rd: 002210
4th: 1010002210
5th: 221022101010002210
The 1-limiting word is the limit of the words for which the number of iterations is congruent to 1 mod 3.
Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
U = 2.28537528186132044169516884721360670506...,
V = 3.87512979416277882597397059430967806752...,
W = 3.28537528186132044169516884721360670506...
If n >=2, then u(n) - u(n-1) is in {1,2,4}, v(n) - v(n-1) is in {2,4,6}, and w(n) - w(n-1) is in {1,3,5,9}.

Examples

			1st iterate: 10
4th iterate: 1010002210

Links

Crossrefs

Cf. A287175, A287332, A287333, A287334.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 2}, 2 -> 0}] &, {0}, 10] (* A287331 *)
Flatten[Position[s, 0]] (* A287332 *)
Flatten[Position[s, 1]] (* A287333 *)
Flatten[Position[s, 2]] (* A287334 *)

Extensions

Definition corrected by Georg Fischer, May 27 2021
Showing 1-8 of 8 results.

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