A334577 -id:A334577 - cp's OEIS frontend

A334576 a(n) is the X-coordinate of the n-th point of the space filling curve P defined in Comments section; sequence A334577 gives Y-coordinates.

0, 1, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6, 5, 4, 4, 4, 4, 5, 6, 6, 6, 7, 7, 7, 7, 6, 5, 5, 6, 7, 7, 7, 8, 9, 10, 10, 10, 11, 11, 11, 12, 13, 14, 14, 13, 12, 12, 12, 11, 10, 9, 9, 9, 8, 8, 8, 8, 9, 10, 10, 9, 8, 8, 8, 8, 9, 10, 10, 10, 11, 11, 11, 12, 13, 14, 14, 13

Offset: 0

Rémy Sigrist, May 06 2020

nonn

The space filling curve P corresponds to the midpoint curve of the alternate paperfolding curve and can be built as follows:
- we define the family {P_k, k > 0}:
    - P_1 corresponds to the points (0, 0), (1, 0), (2, 0) and (2, 1), in that order:
                    +
                    |
                    |
          +----+----+
         O
    - for any k > 0, P_{n+1} is built from four copies of P_n as follows:
                                            +
                                            |A
                +                           |
               C|                   +----+  |
         A     B|    --->           |C  B|  |B  C
        +-------+                   +    |  +----+-+
       O                           C|    |        C|
                             A     B|   A|  A     B|
                            +-------+    +-+-------+
                           O
- the space filling curve P is the limit of P_k as k tends to infinity.
We can also describe the space filling curve P by mean of an L-system (see Links section).

			The first points of the space filling curve P are as follows:
      6|                                  20...21
       |                                  |    |
      5|                                  19   22
       |                                  |    |
      4|                        16...17...18   23
       |                        |              |
      3|                        15   26...25...24
       |                        |    |
      2|              4....5    14   27...28...29
       |              |    |    |              |
      1|              3    6    13...12...11   30
       |              |    |              |    |
      0|    0....1....2    7....8....9....10   31..
       |
    ---+----------------------------------------
    y/x|    0    1    2    3    4    5    6    7
- hence a(9) = a(12) = a(17) = a(26) = a(27) = 5.

Rémy Sigrist, Table of n, a(n) for n = 0..4095
Joerg Arndt, L-system corresponding to P
Robert Ferréol (MathCurve), Courbe de Polya [in French]
Kevin Ryde, Iterations of the Alternate Paperfolding Curve
Rémy Sigrist, Colored line plot of the first 2^14 points of the space filling curve P (where the hue is function of the number of steps from the origin)
Rémy Sigrist, Colored scatterplot of the first 2^20 points of the space filling curve P (where the hue is function of the number of steps from the origin)
Rémy Sigrist, PARI program for A334576
Index entries for sequences related to coordinates of 2D curves

Cf. A020986, A334577.

PARI
```
See Links section.
```

a(n+1) = (A020986(n) + A020986(n+1) - 1)/2 for any n >= 0.

cp's OEIS Frontend

A334576 a(n) is the X-coordinate of the n-th point of the space filling curve P defined in Comments section; sequence A334577 gives Y-coordinates.

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