A294180 The 3-symbol Pell word.

1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 1, 2

Offset: 1

Author

Michel Dekking, Feb 11 2018

Keywords

• nonn

Comments

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

Links

• Michel Dekking, Table of n, a(n) for n = 1..1000
• L. Carlitz, R. Scoville and V. E. Hoggatt, Jr.,Pellian Representations, Fib. Quart. Vol. 10, No. 5, (1972), pp. 449-488.
• F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

Crossrefs

Cf. A270788, A263844, A188376.

Programs

Magma

[Floor((n+2)*r)+Floor((n+1)*r)-2*Floor(n*r)+1 where r is 1-1/Sqrt(2): n in [1..90]]; // Vincenzo Librandi, Feb 23 2018

Maple

a:=[seq(floor((n+2)*(1-1/sqrt(2)))+floor((n+1)*(1-1/sqrt(2)))-2*floor(n*(1-1/sqrt(2)))+1, n=1..130)];

Mathematica

With[{r = 1 - 1/Sqrt[2]}, Table[Inner[Times, Map[Floor[(n + #) r] &, Range[0, 2]], {-2, 1, 1}, Plus] + 1, {n, 108}]] (* Michael De Vlieger, Feb 15 2018 *)

Formula

a(n) = floor((n+2)r)+floor((n+1)r)-2*floor(nr)+1, where r = 1-1/sqrt(2).