A263844 Constant term in expansion of n in Fraenkel's exotic ternary representation.

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

Offset

1,3

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_0.

From Michel Dekking, Feb 11 2018: (Start)

(a(n)) is the unique fixed point of the morphism alpha given by

alpha: 0 -> 012, 1 -> 012, 2 -> 0.

To see this, note first that the p_i satisfy p_{i+2}=2p_{i+1}+p_i for all i=1,2,... Then define a sequence of words by

w(0) = 0, w(1) = 012, w(i+2) = w(i+1)w(i+1)w(i).

The length of w(i) is equal to p_i. In the numeration system, the representation of n = p_i is d = 10..0, and the representation of n = 2p_i is d = 20..0. By unicity of the representation, the numbers n = p_i +m have the representation d = 1c, where c is the representation of m for m = 1,...,p_{i-1}. Similarly, because the digit 2 is required to be followed by the digit 0, the numbers n = 2p_i + m have the representation d =20c, where c is the representation of m for m = 1,...,p_{i-2}. It follows from this that the d_0 digits in the range 0 to p_{i+2} have to satisfy the equation w(i+2) = w(i+1)w(i+1)w(i). But alpha(0)=alpha(1)=w(1), and alpha(2)=w(0), which implies by induction that w(i) = alpha^i(0):

w(i+1) = w(i)w(i)w(i-1) = alpha^i(0) alpha^i(0) alpha^{i-1}(0) =

alpha^i(0) alpha^i(1)alpha^i(2) = alpha^i(012) = alpha^{i+1}(0).

(a(n)) is modulo a change of alphabet (0->1, 1->2, 2->3) equal to A294180, the standard form of (a(n)). This combined with the fact that the Pell word A171588 is a Sturmian word leads to the formula for (a(n)) below. (End)

Links

Michel Dekking, Table of n, a(n) for n = 1..5000

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).)

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

Formula

a(n) = floor((n+2)r) + floor((n+1)r) - 2*floor(nr), where r = 1 - 1/sqrt(2). - Michel Dekking, Feb 11 2018

Example

See the link to Table 2 of Fraenkel (2000).

Mathematica

Table[Floor[(n + 2) (1 - 1/Sqrt[2])] + Floor[(n + 1) (1 - 1/Sqrt[2])] - 2 Floor[n (1 - 1/Sqrt[2])], {n, 100}] (* Vincenzo Librandi, Feb 12 2018 *)

Prog

(Magma) [Floor((n+2)*(1-1/Sqrt(2)))+Floor((n+1)*(1-1/Sqrt(2)))- 2*Floor(n*(1-1/Sqrt(2))): n in [1..100]]; // Vincenzo Librandi, Feb 12 2018

Crossrefs

Cf. A001333, A270788.

Sequence in context: A322989 A322823 A334872 * A079644 A072705 A072574

Adjacent sequences: A263841 A263842 A263843 * A263845 A263846 A263847

Keyword

nonn

Author

N. J. A. Sloane, Nov 06 2015

Extensions

More terms and new offset from Michel Dekking, Feb 11 2018

Status

approved