A287073 -id:A287073 - cp's OEIS frontend

A287072 Start with 0 and repeatedly substitute 0->01, 1->21, 2->0.

0, 1, 2, 1, 0, 2, 1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 2, 1, 0, 1, 2, 1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 2, 1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 2, 1, 0, 1, 2, 1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 2, 1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 1, 0, 2, 1

Offset: 1

Clark Kimberling, May 21 2017

A fixed point of the morphism 0->01, 1->21, 2->0.  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....
Since the morphism 0->01, 1->21, 2->0 is the time reversal of the morphism 0->10, 1->12  2->0, which has fixed point A287104, in particular the incidence matrices of these two morphisms are equal. Thus the algebraic expressions found for U, V and W in A287104 do also apply to the U, V and W above. - Michel Dekking, Sep 15 2019
If n >=2, then u(n) - u(n-1) is in {2,3,4}, v(n) - v(n-1) is in {2,3}, and w(n) - w(n-1) is in {3,4,5}.

Clark Kimberling, Table of n, a(n) for n = 1..10000
James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Properties of a Ternary Infinite Word, arXiv:2206.01776 [cs.DM], 2022.
James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Complement Avoidance in Binary Words, arXiv:2209.09598 [math.CO], 2022.
Index entries for sequences that are fixed points of mappings

Cf. A057985, A287073, A287074, A287075, A287104.

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287072 *)
Flatten[Position[s, 0]] (* A287073 *)
Flatten[Position[s, 1]] (* A287074 *)
Flatten[Position[s, 2]] (* A287075 *)
SubstitutionSystem[{0->{0,1},1->{2,1},2->{0}},{0},{8}][[1]] (* Harvey P. Dale, Feb 18 2025 *)

A287074 Positions of 1 in A287072.

Original entry on oeis.org

2, 4, 7, 9, 12, 14, 16, 18, 21, 23, 25, 28, 30, 32, 34, 37, 39, 41, 44, 46, 49, 51, 53, 56, 58, 60, 62, 65, 67, 69, 72, 74, 77, 79, 81, 83, 86, 88, 90, 93, 95, 98, 100, 102, 105, 107, 109, 111, 114, 116, 118, 121, 123, 126, 128, 130, 132, 135, 137, 139, 142

Offset: 1

Clark Kimberling, May 21 2017

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

Cf. A287072, A287073, A287075.

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287072 *)
Flatten[Position[s, 0]] (* A287073 *)
Flatten[Position[s, 1]] (* A287074 *)
Flatten[Position[s, 2]] (* A287075 *)

A287075 Positions of 2 in A287072.

Original entry on oeis.org

3, 6, 11, 15, 20, 24, 27, 31, 36, 40, 43, 48, 52, 55, 59, 64, 68, 71, 76, 80, 85, 89, 92, 97, 101, 104, 108, 113, 117, 120, 125, 129, 134, 138, 141, 145, 150, 154, 157, 162, 166, 171, 175, 178, 183, 187, 190, 194, 199, 203, 206, 211, 215, 220, 224, 227, 231

Offset: 1

Clark Kimberling, May 21 2017

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

Cf. A287072, A287073, A287074.

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287072 *)
Flatten[Position[s, 0]] (* A287073 *)
Flatten[Position[s, 1]] (* A287074 *)
Flatten[Position[s, 2]] (* A287075 *)

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A287072 Start with 0 and repeatedly substitute 0->01, 1->21, 2->0.

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A287074 Positions of 1 in A287072.

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A287075 Positions of 2 in A287072.

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