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 1-5 of 5 results.

A287175 Positions of 0 in A287174.

Original entry on oeis.org

2, 4, 5, 7, 9, 11, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 31, 33, 35, 37, 38, 40, 42, 44, 46, 48, 49, 51, 53, 55, 57, 59, 61, 62, 64, 66, 68, 70, 72, 74, 75, 77, 79, 81, 83, 85, 86, 88, 90, 92, 94, 96, 98, 99, 101, 103, 105, 106, 108, 110, 112, 114, 116
Offset: 1

Views

Author

Clark Kimberling, May 22 2017

Keywords

Links

Crossrefs

Cf. A287174, 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 *)

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]

A287112 1-limiting word of the morphism 0->10, 1->20, 2->0.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 22 2017

Keywords

Comments

Starting with 0, the first 4 iterations of the morphism yield words shown here:
1st: 10
2nd: 2010
3rd: 0102010
4th: 1020100102010
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 = 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}.
From Michel Dekking, Mar 29 2019: (Start)
This sequence is one of the three fixed points of the morphism alpha^3, where alpha is the defining morphism
0->10, 1->20, 2->0.
The other two fixed points are A286998 and A287174.
We have alpha = rho(tau), where tau is the tribonacci morphism in A080843
0->01, 1->02, 2->0,
and rho is the rotation operator.
An eigenvector computation of the incidence matrix of the morphism gives that 0,1, and 2 have frequencies 1/t, 1/t^2 and 1/t^3, where t is the tribonacci constant A058265.
Apparently (u(n)) := A287113. Thus U, the limit of u(n)/n, equals 1/(1/t), the tribonacci constant t. Also, V = A276800, and W = A276801.
(End)

Examples

			1st iterate: 10
4th iterate: 1020100102010
7th iterate:  102010010201020100102010102010010201001020101020100102010201001020101020100102010.

Links

Crossrefs

Cf. A287113, A287114, A287115, A286998, A287174.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 10]   (* A287112 *)
Flatten[Position[s, 0]] (* A287113 *)
Flatten[Position[s, 1]] (* A287114 *)
Flatten[Position[s, 2]] (* A287115 *)
Showing 1-5 of 5 results.

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