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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

From Michel Dekking, Sep 17 2019: (Start)

Let sigma be the defining morphism of A287104: 0->10, 1->12, 2->0.

Let u=01, v=012, w=0121 be the return words of the word 0.

[See Justin & Vuillon (2000) for definition of return word. - N. J. A. Sloane, Sep 23 2019]

Then under sigma u, v and w are mapped to

sigma(01) = 1012, sigma(012) = 10120, sigma(0121) = 1012012.

Moving the prefix 1 of these three images to the end, the sequence 0 a (i.e., (a(n)) prefixed by the symbol 0), is a fixed point when iterating.

This iteration process induces a morphism 2->4, 3->32, 4->34 on the return words, coded by their lengths.

Coding the symbols according to 2<->2, 4<->0, 3<->1, this leads to the morphism 2->0, 1->12, 0->10 on the alphabet {0,1,2}.

This is simply sigma, which has A287104 as its unique fixed point. So the sequence d of first differences of (a(n)) equals A287104 with the coding above. Noting that the code can be written as x->4-x, this gives the formula below.

(End)

From Michel Dekking, Sep 16 2019: (Start)

Let sigma be the defining morphism of A287104: 0->10, 1->12, 2->0.

Let u=201, v=2101, w=20101 be the return words of the word 2.

Under sigma u, v, and w are mapped to sigma(201) = 01012, sigma(2101) = 0121012, sigma(20101) = 010121012.

All three images have suffix 2. We can therefore move this suffix to the front of all three images, obtaining the fixed point (a(n+1)) = 20101... when iterating. This induces the morphism 3 -> 5, 4 -> 34, 5 -> 54 on the return words, coded by their lengths.

Coding the symbols according to 3<->2, 4<->1, 5<->0, this leads to the morphism 2->0, 1->21, 0->01 on the alphabet {0,1,2}. This is exactly the morphism which has A287072 as unique fixed point. So the sequence d of first differences of (a(n)) equals A287072 with the coding above. This gives the formula below.

(End)

Starting with 0, the first 4 iterations of the morphism yield words shown here:

1st: 10

2nd: 2110

3rd: 0212110

4th: 1002010212110

The 0-limiting word is the limit of the words for which the number of iterations is congruent to 0 mod 3.

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....

If n >=2, then u(n) - u(n-1) is in {1,2,3,4,6}, v(n) - v(n-1) is in {1,2,3,4}, and w(n) - w(n-1) is in {2,3,4,5,7}.