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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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).

LINKS

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

FORMULA

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

EXAMPLE

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.

PROG

(PARI) See Links section.

CROSSREFS

Cf. A020990, A334576.

Sequence in context: A255124 A030619 A016370 * A219494 A334221 A089069

Adjacent sequences:  A334574 A334575 A334576 * A334578 A334579 A334580

KEYWORD

nonn

AUTHOR

Rémy Sigrist, May 06 2020

STATUS

approved

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Last modified April 22 22:19 EDT 2021. Contains 343197 sequences. (Running on oeis4.)