A332299 -id:A332299 - cp's OEIS frontend

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

Rémy Sigrist, Feb 09 2020

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.

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

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

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) }

A265318 Fibonacci word fractal in an n X n grid, starting downwards from the top-left corner, listed antidiagonally.

Original entry on oeis.org

1, 0, 2, 5, 3, 0, 6, 4, 0, 0, 7, 0, 0, 0, 0, 0, 8, 10, 0, 0, 20, 0, 0, 9, 11, 0, 19, 21, 0, 0, 0, 0, 12, 18, 0, 22, 0, 0, 0, 0, 13, 0, 17, 23, 0, 0, 0, 0, 0, 0, 14, 16, 0, 24, 26, 0, 0, 0, 0, 0, 0, 15, 0, 0, 25, 27, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 28, 83

Offset: 1

Max Barrentine, Dec 06 2015

The n-th iteration of this curve ends at the n-th Fibonacci number.

As this is not a space-filling curve, not all points on the grid are reached by the curve; these points are given the value 0.

			The top left corner of the array shows how this curve begins (connect the terms in numerical order):
   1   0   5   6   7
   2   3   4   0   8
   0   0   0  10   9
   0   0   0  11   0
   0   0   0  12  13
  20  19  18   0  14
  21   0  17  16  15
  22  23   0   0   0
   0  24   0   0   0
  26  25   0   0   0
  27   0  31  32  33
  28  29  30   0  34

Max Barrentine, Table of n, a(n) for n = 1..2484
Alexis Monnerot-Dumaine, The Fibonacci Word fractal, HAL Id: hal-00367972, 2009.
Alexis Monnerot-Dumaine, The Fibonacci word fractal [Cached copy, with permission]

Cf. A332298, A332299.

See also A163357, A163334, and A054238 for other fractal curves.

cp's OEIS Frontend

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

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