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

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, 7, 7, 6, 6, 5, 5, 4, 4, 5, 5, 6, 6, 7, 7, 6, 6, 7, 7, 8, 8, 9, 9, 8, 8, 9, 9, 10, 10, 11, 11, 10, 10, 9, 9, 8, 8, 9, 9, 8, 8, 9, 9, 10, 10, 11, 11, 10, 10, 11, 11, 12, 12, 13, 13, 12

Offset

0,4

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.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..5461

Robert Ferréol (MathCurve), Courbe de Koch quadratique [in French]

Rémy Sigrist, Line plot of the first 5462 points

Rémy Sigrist, PARI program for A340327

Index entries for sequences related to coordinates of 2D curves

Example

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(0) = 0,
     a(5) = a(6) = a(9) = a(10) = 3.

Prog

(PARI) See Links section.

Crossrefs

See A332249 and A340320 for similar sequences.

Cf. A340328 (Y-coordinates).

Sequence in context: A287271 A290884 A332383 * A332249 A049113 A055093

Adjacent sequences: A340324 A340325 A340326 * A340328 A340329 A340330

Keyword

nonn

Author

Rémy Sigrist, Jan 04 2021

Status

approved