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 1-4 of 4 results.

A287179 1-limiting word of the morphism 0->10, 1->20, 2->1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 1
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: 1102010
4th: 2020101102010
5th: 11011020102020101102010
The 1-limiting word is the limit of the words for which the number of iterations is odd.
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 = 2.246979603717467061050009768008...,
V = 2.801937735804838252472204639014...,
W = 5.048917339522305313522214407023...
If n >=2, then u(n) - u(n-1) is in {2,3}, v(n) - v(n-1) is in {1,2,4,6}, and w(n) - w(n-1) is in {2,4,7,10}.

Examples

			1st iterate: 10
3rd iterate: 1102010
5th iterate: 110110201020201011020100

Links

Crossrefs

Cf. A287121, A287180, A287181, A287182.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 1}] &, {0}, 9] (* A287179 *)
Flatten[Position[s, 0]] (* A287180 *)
Flatten[Position[s, 1]] (* A287181 *)
Flatten[Position[s, 2]] (* A287182 *)

A287180 Positions of 0 in A287179.

Original entry on oeis.org

3, 6, 8, 10, 13, 16, 18, 20, 22, 24, 26, 29, 31, 33, 36, 39, 41, 43, 46, 49, 51, 53, 55, 57, 59, 62, 64, 66, 68, 70, 72, 74, 76, 78, 81, 83, 85, 88, 91, 93, 95, 97, 99, 101, 104, 106, 108, 111, 114, 116, 118, 121, 124, 126, 128, 130, 132, 134, 137, 139, 141
Offset: 1

Views

Author

Clark Kimberling, May 22 2017

Keywords

Links

Crossrefs

Cf. A287179, A287181, A287182.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 1}] &, {0}, 9] (* A287179 *)
Flatten[Position[s, 0]] (* A287180 *)
Flatten[Position[s, 1]] (* A287181 *)
Flatten[Position[s, 2]] (* A287182 *)

A287182 Positions of 2 in A287179.

Original entry on oeis.org

7, 17, 21, 23, 30, 40, 50, 54, 56, 63, 67, 69, 73, 75, 82, 92, 96, 98, 105, 115, 125, 129, 131, 138, 148, 158, 162, 164, 171, 175, 177, 181, 183, 190, 200, 204, 206, 213, 217, 219, 223, 225, 232, 236, 238, 242, 244, 251, 261, 265, 267, 274, 284, 294, 298
Offset: 1

Views

Author

Clark Kimberling, May 23 2017

Keywords

Links

Crossrefs

Cf. A287179, A287180, A287181.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 1}] &, {0}, 9] (* A287179 *)
Flatten[Position[s, 0]] (* A287180 *)
Flatten[Position[s, 1]] (* A287181 *)
Flatten[Position[s, 2]] (* A287182 *)

A287320 0-limiting word of the morphism 0->10, 1->22, 2->0.

Original entry on oeis.org

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, 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, 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
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 0-limiting word is the limit of the words for which the number of iterations is congruent to 0 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

			3rd iterate: 002210
6th iterate: 002210002210221022101010002210

Links

Crossrefs

Cf. A287321, A287322, A287323, A287181. A298200.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 2}, 2 -> 0}] &, {0}, 12] (* A287320 *)
Flatten[Position[s, 0]] (* A287321 *)
Flatten[Position[s, 1]] (* A287322 *)
Flatten[Position[s, 2]] (* A287323 *)
SubstitutionSystem[{0->{1,0},1->{2,2},2->{0}},{2},{10}][[1]] (* Harvey P. Dale, Aug 17 2022 *)
Showing 1-4 of 4 results.

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