A340321 a(n) is the Y-coordinate of the n-th point of a variant of the quadratic Koch curve. Sequence A340320 gives X-coordinates.

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

Offset: 0

Author

Rémy Sigrist, Jan 04 2021

Keywords

• nonn

Comments

The curve is built by successively applying the following substitution to an initial vector (1, 0) (the two vertical copies are horizontally flipped):

*

.------>.

^ |

|* *|

* | v *

.------>. .------>.

The quadratic Koch curve is built without horizontal flip.

Examples

			The curve starts as follows:
                     +---+
                     |12 |13
                     |   |
                 +---+   +---+
                 |10  11  14 |15
                 |           |
                 +---+   +---+
                  9  |8  |17  16
                     |   |
         +---+   +---+   +---+   +---+
         |2  |3  |6   7   18 |19 |22 |23
         |   |   |           |   |   |
     +---+   +---+           +---+   +---+
      0   1   4   5           20  21  24  25
- so a(0) = a(1) = a(4) = a(5) = a(20) = a(21) = a(24) = a(25) = 0,
     a(8) = a(9) = a(16) = a(17) = 2.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..3125
• Robert Ferréol (MathCurve), Courbe de Koch quadratique [in French]
• Rémy Sigrist, PARI program for A340321
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

See A332250 and A340328 for similar sequences.

Cf. A340320 (X-coordinates).

Programs

PARI

See Links section.

Formula

a(5^k-m) = a(m) for any k >= 0 and m = 0..5^k.