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 1-3 of 3 results.

A286998 0-limiting word of the morphism 0->10, 1->20, 2->0.

Original entry on oeis.org

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, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 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..., (A058265)
V = U^2 = 3.3829757679062374941227085364..., (A276800)
W = U^3 = 6.2222625231203986266745611011.... (A276801)
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}.
From Jiri Hladky, Aug 29 2021: (Start)
This is also Arnoux-Rauzy word sigma_0 x sigma_1 x sigma_2, where sigmas are defined as:
sigma_0 : 0 -> 0, 1 -> 10, 2 -> 20;
sigma_1 : 0 -> 01, 1 -> 1, 2 -> 21;
sigma_2 : 0 -> 02, 1 -> 12, 2 -> 2.
Fixed point of the morphism 0->0102010, 1->102010, 2->2010, starting from a(1)=0. This definition has the benefit that EACH iteration yields the prefix of the limiting word.
Frequency of letters:
0: 1/t ~ 54.368% (A192918)
1: 1/t^2 ~ 29.559%
2: 1/t^3 ~ 16.071%
where t is tribonacci constant A058265.
Equals A347290 with a re-mapping of values 1->2, 2->1.
(End)

Examples

			3rd iterate: 0102010
6th iterate: 01020101020100102010201001020101020100102010

Links

Crossrefs

Cf. A080843, A286999, A287000, A287001, A287112, A287174, A347290 (values 0,2,1).

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 9] (* A286998 *)
Flatten[Position[s, 0]] (* A286999 *)
Flatten[Position[s, 1]] (* A287000 *)
Flatten[Position[s, 2]] (* A287001 *)
(* Using the 0->0102010, 1->102010, 2->2010 rule: *)
Nest[ Flatten[# /. {0 -> {0, 1, 0, 2, 0, 1, 0}, 1 -> {1, 0, 2, 0, 1, 0}, 2 -> {2, 0, 1, 0}}] &, {0}, 3]

A287000 Positions of 1 in A286998.

Original entry on oeis.org

2, 6, 8, 12, 15, 19, 23, 26, 30, 32, 36, 39, 43, 45, 49, 52, 56, 60, 63, 67, 69, 73, 76, 80, 83, 87, 89, 93, 96, 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

Links

Crossrefs

Cf. A286998, A286999, A287001.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 9] (* A286998 *)
Flatten[Position[s, 0]] (* A286999 *)
Flatten[Position[s, 1]] (* A287000 *)
Flatten[Position[s, 2]] (* A287001 *)

A287001 Positions of 2 in A286998.

Original entry on oeis.org

4, 10, 17, 21, 28, 34, 41, 47, 54, 58, 65, 71, 78, 85, 91, 98, 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, 277, 284, 288, 295, 301, 308, 315, 321, 328, 332, 339
Offset: 1

Views

Author

Clark Kimberling, May 22 2017

Keywords

Links

Crossrefs

Cf. A286998, A286999, A287000.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 9] (* A286998 *)
Flatten[Position[s, 0]] (* A286999 *)
Flatten[Position[s, 1]] (* A287000 *)
Flatten[Position[s, 2]] (* A287001 *)
Showing 1-3 of 3 results.

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