A307247 Second digit in the expansion of n in Fraenkel's exotic ternary representation.

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

Offset: 1

Author

Michel Dekking, Apr 01 2019

Keywords

• nonn

Comments

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.

Links

• Michel Dekking, Table of n, a(n) for n = 1..5000 (restored by _Georg Fischer_, Apr 05 2019)
• F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
• Aviezri S. Fraenkel, On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications, Discrete Mathematics 224 (2000), pp. 273-279.
• A. S. Fraenkel, An exotic ternary representation of the first few positive integers (Table 2 from Fraenkel (2000).)

Crossrefs

Cf. A263844, A001333.