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.

A287108 1-limiting word of the morphism 0->10, 1->21, 2->0.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 21 2017

Keywords

Comments

Starting with 0, the first 4 iterations of the morphism yield words shown here:
1st: 10
2nd: 2110
3rd: 0212110
4th: 100210212110
The 1-limiting word is the limit of the words for which the number of iterations is congruent to 1 mod 3.
Let u be the sequence of positions of 0, and likewise, v for 1 and w for 2. 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 = 3.079595623491438786010417...,
V = 2.324717957244746025960908...,
W = U + 1 = 4.079595623491438786010417....
If n >=2, then u(n) - u(n-1) is in {1,2,3,4,6}, v(n) - v(n-1) is in {1,2,3,4}, and w(n) - w(n-1) is in {2,3,4,5,7}.

Examples

			The 1st, 4th, and 7th iterates are
10, 100210212110, 10021021211002121102110100210212110211010021211010021100210212110.

Links

Crossrefs

Cf. A287109, A287110, A287111.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287108 *)
Flatten[Position[s, 0]] (* A287109 *)
Flatten[Position[s, 1]] (* A287110 *)
Flatten[Position[s, 2]] (* A287111 *)

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 *)

A287109 Positions of 0 in A287108.

Original entry on oeis.org

2, 3, 6, 12, 13, 19, 23, 25, 26, 29, 35, 39, 41, 42, 48, 50, 51, 55, 56, 59, 65, 66, 72, 76, 78, 79, 85, 87, 88, 92, 93, 96, 102, 106, 108, 109, 113, 114, 117, 123, 125, 126, 129, 135, 136, 142, 146, 148, 149, 152, 158, 162, 164, 165, 171, 173, 174, 178, 179
Offset: 1

Views

Author

Clark Kimberling, May 21 2017

Keywords

Links

Crossrefs

Cf. A287108, A287110, A287111.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287108 *)
Flatten[Position[s, 0]] (* A287109 *)
Flatten[Position[s, 1]] (* A287110 *)
Flatten[Position[s, 2]] (* A287111 *)

A287110 Positions of 1 in A287108.

Original entry on oeis.org

1, 5, 8, 10, 11, 15, 17, 18, 21, 22, 24, 28, 31, 33, 34, 37, 38, 40, 44, 46, 47, 49, 53, 54, 58, 61, 63, 64, 68, 70, 71, 74, 75, 77, 81, 83, 84, 86, 90, 91, 95, 98, 100, 101, 104, 105, 107, 111, 112, 116, 119, 121, 122, 124, 128, 131, 133, 134, 138, 140, 141
Offset: 1

Views

Author

Clark Kimberling, May 21 2017

Keywords

Links

Crossrefs

Cf. A287108, A287109, A287111.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287108 *)
Flatten[Position[s, 0]] (* A287109 *)
Flatten[Position[s, 1]] (* A287110 *)
Flatten[Position[s, 2]] (* A287111 *)
Showing 1-4 of 4 results.

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