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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, Jan 04 2021

Keywords

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

Links

Crossrefs

See A332249 and A340320 for similar sequences.
Cf. A340328 (Y-coordinates).

Programs

  • PARI
    See Links section.

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.

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, Jan 04 2021

Keywords

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

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.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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