A332249 -id:A332249 - cp's OEIS frontend

A332250 a(n) is the Y-coordinate of the n-th point of the quadratic Koch curve. Sequence A332249 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, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 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

Rémy Sigrist, Feb 08 2020

This sequence is the imaginary part of {f(n)} defined as:
- f(0) = 0,
- f(n+1) = f(n) + i^t(n)
  where t(n) is the number of 1's minus the number of 3's
             in the base 5 representation of n
    and i denotes the imaginary unit.

Rémy Sigrist, Table of n, a(n) for n = 0..15625
Index entries for sequences related to coordinates of 2D curves

Cf. A332249 (X-coordinates and additional comments).

{ k = [0, 1, 0, -1, 0]; z=0; for (n=0, 80, print1 (imag(z) ", "); z += I^vecsum(apply(d -> k[1+d], digits(n, #k)))) }

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

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
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See Links section.
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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

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

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

See A332249 and A340320 for similar sequences.

Cf. A340328 (Y-coordinates).

PARI
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See Links section.
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A229217 If 1 and 2 represent the 2D vectors (1,0) and (0,1) and -1 and -2 are the negation of these vectors, then this sequence represents the Koch curve.

Original entry on oeis.org

1, 2, 1, -2, 1, 2, -1, 2, 1, 2, 1, 2, 1, -2, 1, -2, 1, -2, -1, -2, 1, 2, 1, -2, 1, 2, -1, 2, 1, 2, -1, -2, -1, 2, -1, 2, -1, 2, 1, 2, 1, 2, 1, -2, 1, 2, -1, 2, 1, 2, 1, 2, 1, -2, 1, 2, -1, 2, 1, 2, 1, 2, 1, -2, 1, -2, 1, -2, -1, -2, 1, 2, 1, -2, 1, -2, 1, -2, -1, -2, 1, 2, 1, -2, 1, -2, 1, -2, -1, -2, -1, -2, -1, 2, -1, -2, 1, -2, -1, -2, 1, 2, 1, -2, 1, 2, -1

Offset: 1

Arie Bos, Sep 25 2013

sign

The sequence is generated by the rewriting rules:
P(1) = 1,2,1,-2,1;
P(2) = 2,-1,2,1,2 and
P(-1) = -1,-2,-1,2,-1;
P(-2) = -2,1,-2,-1,-2, so P(-x)=-P(x).
The start is 1.

			Start with 1, you get
in the first step 1,2,1,-2,1, and
in the 2nd step 1,2,1,-2,1,2,-1,2,1,2,1,2,1,-2,1,-2,1,-2,-1,-2,1,2,1,-2,1.
With each step the length increases by a factor 5.

Arie Bos, Index notation of grid graphs, arXiv:1210.7123 [cs.CG], 2012.
Wikipedia, Koch curve
Index entries for sequences that are fixed points of mappings

Coordinates: A332249, A332250.

Cf. A229216, A166253.

cp's OEIS Frontend

A332250 a(n) is the Y-coordinate of the n-th point of the quadratic Koch curve. Sequence A332249 gives X-coordinates.

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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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A229217 If 1 and 2 represent the 2D vectors (1,0) and (0,1) and -1 and -2 are the negation of these vectors, then this sequence represents the Koch curve.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs