A340327 -id:A340327 - cp's OEIS frontend

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

Rémy Sigrist, Jan 04 2021

nonn

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.

			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.

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

See A332250 and A340321 for similar sequences.

Cf. A340327 (X-coordinates).

PARI
```
See Links section.
```

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

Original entry on oeis.org

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, 10, 10, 11, 11, 12, 12, 13, 13, 12, 12, 13, 13, 14, 14, 15, 15, 14, 14, 15, 15, 16, 16

Offset: 0

Rémy Sigrist, Jan 04 2021

nonn

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.

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

Rémy Sigrist, Table of n, a(n) for n = 0..3125
Robert Ferréol (MathCurve), Courbe de Koch quadratique [in French]
Rémy Sigrist, Line plot of the first 1+5^5 points
Rémy Sigrist, PARI program for A340320
Index entries for sequences related to coordinates of 2D curves

See A332249 and A340327 for similar sequences.

Cf. A340321 (Y-coordinates).

PARI
```
See Links section.
```

cp's OEIS Frontend

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.

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