cp's OEIS Frontend

In the Pell word A171588 = 0, 0, 1, 0, 0, 1, 0, 0, 0, ..., group the letters in overlapping blocks of length two: [0,0],[0,1],[1,0],[0,0],[0,1],[1,0],... Then code [0,0]->1, [0,1]->2, [1,0]->3. This gives (a(n)).

(a(n)) is the unique fixed point of the 3-symbol Pell morphism

1 -> 123, 2 ->123, 3 -> 1.

The morphism and the fixed point are in standard form.

Modulo a change of alphabet (1->0, 2->1, 3->2), this sequence is equal to A263844.

From Michel Dekking, Feb 23 2018: (Start)

The positions of 1 in (a(n)) are given by

A188376 = 1,4,7,8,11,14,15,18,...

Why is this true? First, the Pell word b is given by

b(n) = [(n+1)(1-r)]-[n(1-r)], where r =1/sqrt(2).

This can rewritten as

b(n) = [nr]-[(n+1)r]+1.

Second,

1 occurs at n in (a(n)) <=>

00 occurs at n in (b(n)) <=>

b(n)+b(n+1) = 0 <=>

[nr]-[(n+2)r]+2 = 0 <=>

[(n+2)r]-[nr]-1 = 1 <=>

1 occurs at n in A188374.

The positions of 2 in (a(n)) are given by A001952 - 1 = 2,5,9,12,16,..., since 2 occurs at n in (a(n)) if and only if 3 occurs at n+1 in (a(n)).

The positions of 3 in (a(n)) are given by A001952 = 3,6,10,13,17,..., since 3 occurs at n in (a(n)) if and only if 1 occurs at n in (b(n)).

The sequence of positions of 3 in (a(n)) is equal to the sequence b in Carlitz et al. The sequence of positions of 1 in (a(n)) seems to be equal to the sequence ad' in Carlitz et al. (End)

See the comments of A188376 for a proof of the observation on the positions of 1 in (a(n)). - Michel Dekking, Feb 27 2018

Let {p_i, i >= 0} = {1,3,7,17,41,99,...} denote the numerators of successive convergents to sqrt(2) (see A001333). Then any n >= 0 has a unique representation as n = Sum_{i >= 0} d_i*p_i, with 0 <= d_i <= 2, d_{i+1}=2 => d_i=0. Sequence gives a(n+1) = d_1.

Let x be the 3-symbol Pell word A294180 = 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, ... Let delta be the morphism

1 -> 000, 2 -> 111, 3 -> 2.

Then delta(x) = (a(n)). This can be proved by induction, starting from the knowledge that the sequence of first digits d_0 = d_0(n) of n in the exotic ternary expansion shifted by 1 is equal to x (see A263844).

More generally, the sequence of k-th digits d_k shifted by 1 is equal to delta_k(x), where the morphism delta_k is given by

1 -> U_k, 2 -> V_k, 3 -> W_k.

Here U_k is a concatenation of p_{k+1} letters 0, V_k is a concatenation of p_{k+1} letters 1, and W_k is a concatenation of p_k letters 2.