A332298 a(n) is the X-coordinate of the n-th point of the Fibonacci word fractal. Sequence A332299 gives Y-coordinates.

0, 1, 2, 2, 3, 4, 4, 4, 3, 3, 3, 4, 4, 4, 3, 2, 2, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 6, 5, 5, 5, 6, 7, 7, 8, 9, 9, 10, 11, 11, 11, 10, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 16, 15, 15, 15, 16, 16, 16, 15, 14, 14, 13, 12, 12, 12, 13

Offset: 0

Author

Rémy Sigrist, Feb 09 2020

Keywords

• nonn
• look

Comments

To build the curve:

- start from the origin looking to the right,

- for k=0, 1, ...:

- move forward to the next lattice point,

- if A003849(k) = 1 then:

- if k is even then turn 90 degrees to the right

- otherwise turn 90 degrees to the left.

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10946
• Robert Ferréol (MathCurve), Courbe du mot de Fibonacci [in French]
• Alexis Monnerot-Dumaine, The Fibonacci Word Fractal, HAL-00367972, February 2009.
• Alexis Monnerot-Dumaine, The Fibonacci word fractal [Cached copy, with permission]
• Rémy Sigrist, Representation of the first 1+F(24) points of the Fibonacci word fractal
• Rémy Sigrist, Colored representation of the first 1+F(24) points of the Fibonacci word fractal (where the color is function of the greatest Fibonacci number <= n)
• Wikipedia, Fibonacci word fractal
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

Cf. A003849, A265318, A332299 (Y-coordinates).

Programs

PARI

A003849(n)=my(k=2); while(fibonacci(k)<=n, k++); while(n>1, while(fibonacci(k--)>n, ); n-=fibonacci(k)); n==1
{ z=0; d=I; for (n=0, 76, print1 (real(z) ", "); if (A003849(n)==0, if (n%2==0, d/=I, d*=I);); z+=d) }