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 1-6 of 6 results.

A286999 Positions of 0 in A286998.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 14, 16, 18, 20, 22, 24, 25, 27, 29, 31, 33, 35, 37, 38, 40, 42, 44, 46, 48, 50, 51, 53, 55, 57, 59, 61, 62, 64, 66, 68, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 88, 90, 92, 94, 95, 97, 99, 101, 103, 105, 106, 108, 110, 112, 114, 116
Offset: 1

Views

Author

Clark Kimberling, May 22 2017

Keywords

Links

Crossrefs

Cf. A286998, A287000, 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 *)

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 *)

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 *)

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 *)

A347290 Arnoux-Rauzy word sigma_0 x sigma_2 x sigma_1. Fixed point of the morphism 0-> 0201020, 1->1020, 2->201020 starting from a(1)=0.

Original entry on oeis.org

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

Views

Author

Jiri Hladky, Aug 29 2021

Keywords

Comments

Arnoux-Rauzy word sigma_0 x sigma_2 x sigma_1, 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->0201020, 1->1020, 2->201020 starting from a(1)=0.
Frequency of letters:
0: 1/t ~ 54.368% (A192918)
1: 1/t^3 ~ 16.071%
2: 1/t^2 ~ 29.559%
where t is tribonacci constant A058265.
Equals A286998 with a re-mapping of values 1->2, 2->1.

Links

Crossrefs

Cf. A080843 A286998 (values 0,2,1), A058265.

Programs

  • Mathematica
    Nest[ Flatten[# /. {0 -> {0, 2, 0, 1, 0, 2, 0}, 1 -> {1, 0, 2, 0}, 2 -> {2, 0, 1, 0, 2, 0}}] &, {0}, 3]
Showing 1-6 of 6 results.

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