A334576 -id:A334576 - cp's OEIS frontend

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

0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 4, 4, 5, 6, 6, 5, 4, 3, 3, 3, 2, 2, 2, 1, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 2, 3, 4, 4, 4, 5, 5, 5, 6, 7, 8, 8, 8, 9, 10, 10, 9, 8, 8, 8, 8, 9, 9, 9, 10, 11, 12, 12, 12, 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(15) = a(24) = a(25) = a(26) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..4095
Joerg Arndt, L-system corresponding to P
Kevin Ryde, Iterations of the Alternate Paperfolding Curve
Rémy Sigrist, PARI program for A334577
Index entries for sequences related to coordinates of 2D curves

Cf. A020990, A334576.

PARI
```
See Links section.
```

a(n+1) = (A020990(n) + A020990(n+1) - 1)/2 for any n >= 0.

cp's OEIS Frontend

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

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