A287104 Start with 0 and repeatedly substitute 0->10, 1->12, 2->0.

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

Offset: 1

Author

Clark Kimberling, May 21 2017

Keywords

• nonn
• easy

Comments

The fixed point of the morphism 0->10, 1->12, 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. It appears that 1/U + 1/V + 1/W = 1, where

U = 3.079595623491438786010417...,

V = 2.324717957244746025960908...,

W = U + 1 = 4.079595623491438786010417....

From Michel Dekking, Sep 15 2019: (Start)

The incidence matrix of the morphism sigma: 0->10, 1->12, 2->0 has characteristic polynomial chi(u) = u^3-2u^2+u-1. The real root of chi is lambda := Q/6 + 2/3*1/Q + 2/3, where

Q = ( 100 + 12*sqrt(69) )^1/3.

An eigenvector of lambda is (1, lambda^2-lambda, lambda-1).

The Perron-Frobenius Theorem gives that the asymptotic frequencies f0, f1 and f2 of the letters 0, 1, and 2 are

f0 = 1/lambda^2,

f1 = (lambda^2 - lambda +1)/lambda^3,

f2 = (lambda - 1)/lambda^2.

Algebraic expressions for the constants U,V and W are then given by

U = 1/f0, V = 1/f1, W = 1/f2.

In particular, this shows that W = U + 1.

(End)

Conjecture: 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}.

See A287105, A287106, and A287107 for proofs of these conjectures, with explicit expressions for u, v, and w. - Michel Dekking, Sep 15 2019

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000
• James D. Currie, Words with factor somplexity 2n+1 and minimal critical exponent, arXiv:2507.09387 [math.CO], 2025. See p. 2.
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A287105, A287106, A287107.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 2}, 2 -> 0}] &, {0}, 10] (* A287104 *)
Flatten[Position[s, 0]] (* A287105 *)
Flatten[Position[s, 1]] (* A287106 *)
Flatten[Position[s, 2]] (* A287107 *)