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

Author

Rémy Sigrist, May 06 2020

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

• 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

Crossrefs

Cf. A020986, A334577.

Programs

PARI

See Links section.

Formula

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