cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

N. J. A. Sloane, Nov 06 2015

Keywords

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)

Examples

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

Crossrefs

Programs

  • 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
  • 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 *)

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

Extensions

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