A287115 -id:A287115 - cp's OEIS frontend

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

Clark Kimberling, May 22 2017

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)

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

Clark Kimberling, Table of n, a(n) for n = 1..10000

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

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 *)

A287113 Positions of 0 in A287112.

Original entry on oeis.org

2, 4, 6, 7, 9, 11, 13, 15, 17, 18, 20, 22, 24, 26, 28, 30, 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, 83, 85, 86, 88, 90, 92, 94, 96, 98, 99, 101, 103, 105, 106, 108, 110, 112, 114, 116

Offset: 1

Clark Kimberling, May 22 2017

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A287112, A287114, A287115.

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 *)

A287114 Positions of 1 in A287112.

Original entry on oeis.org

1, 5, 8, 12, 16, 19, 23, 25, 29, 32, 36, 39, 43, 45, 49, 52, 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, 150, 154, 157, 161, 165, 168, 172, 174, 178, 181, 185, 188, 192, 194

Offset: 1

Clark Kimberling, May 22 2017

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A287112, A287113, A287115.

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 *)

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

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A287113 Positions of 0 in A287112.

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A287114 Positions of 1 in A287112.

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