A340328 a(n) is the Y-coordinate of the n-th point of a variant of the quadratic Koch curve. Sequence A340327 gives X-coordinates.

0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 5, 5, 4, 4, 5, 5, 6, 6, 7, 7, 6, 6, 5, 5, 4, 4, 5, 5, 4, 4, 3, 3, 2, 2, 3, 3, 2, 2, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 1, 1, 0, 0, 1, 1, 0, 0, 1

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) (we have 4 copies and a horizontal unit vector):

.-.

^ |

| |

| v

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

The curve visits once or twice every lattice point (x, y) such that 0 <= y <= x.

The quadratic Koch curve is built from 5 copies at each step.

Examples

			The curve starts as follows:
                                    +
                                    |42
                                    |
                               +----+
                               |40   41
                               |
                          +----+----+
                          |34  |35  |38
                          |    |39  |
                     +----+    +----+
                     |32   33   36   37
                     |
                +----+----+    +----+
                |10  |11  |30  |27  |26
                |    |31  |    |    |
           +----+    +----+----+----+
           |8    9    12  |13  |24   25
           |              |29  |28
      +----+----+    +----+----+----+
      |2   |3   |6   |15  |14  |19  |22
      |    |7   |    |    |18  |23  |
+----+    +----+    +----+    +----+
  0    1    4    5    16   17   20   21
- so a(8) = a(9) = a(12) = a(13) = a(24) = a(25) = a(28) = a(29) = 2,
     a(40) = a(41) = 6.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..5461
• Robert Ferréol (MathCurve), Courbe de Koch quadratique [in French]
• Rémy Sigrist, PARI program for A340328
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

See A332250 and A340321 for similar sequences.

Cf. A340327 (X-coordinates).

Programs

PARI

See Links section.