A334576 - OEIS

%I #23 May 09 2020 02:31:55

%S 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,

%T 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,

%U 10,9,8,8,8,8,9,10,10,10,11,11,11,12,13,14,14,13

%N 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.

%C The space filling curve P corresponds to the midpoint curve of the alternate paperfolding curve and can be built as follows:

%C - we define the family {P_k, k > 0}:

%C     - P_1 corresponds to the points (0, 0), (1, 0), (2, 0) and (2, 1), in that order:

%C                     +

%C                     |

%C                     |

%C           +----+----+

%C          O

%C     - for any k > 0, P_{n+1} is built from four copies of P_n as follows:

%C                                             +

%C                                             |A

%C                 +                           |

%C                C|                   +----+  |

%C          A     B|    --->           |C  B|  |B  C

%C         +-------+                   +    |  +----+-+

%C        O                           C|    |        C|

%C                              A     B|   A|  A     B|

%C                             +-------+    +-+-------+

%C                            O

%C - the space filling curve P is the limit of P_k as k tends to infinity.

%C We can also describe the space filling curve P by mean of an L-system (see Links section).

%H Rémy Sigrist, <a href="/A334576/b334576.txt">Table of n, a(n) for n = 0..4095</a>

%H Joerg Arndt, <a href="/A334576/a334576.pdf">L-system corresponding to P</a>

%H Robert Ferréol (MathCurve), <a href="https://mathcurve.com/fractals/polya/polya.shtml">Courbe de Polya</a> [in French]

%H Kevin Ryde, <a href="https://user42.tuxfamily.org/alternate/index.html">Iterations of the Alternate Paperfolding Curve</a>

%H Rémy Sigrist, <a href="/A334576/a334576.png">Colored line plot of the first 2^14 points of the space filling curve P</a> (where the hue is function of the number of steps from the origin)

%H Rémy Sigrist, <a href="/A334576/a334576_1.png">Colored scatterplot of the first 2^20 points of the space filling curve P</a> (where the hue is function of the number of steps from the origin)

%H Rémy Sigrist, <a href="/A334576/a334576.gp.txt">PARI program for A334576</a>

%H <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>

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

%e The first points of the space filling curve P are as follows:

%e       6|                                  20...21

%e        |                                  |    |

%e       5|                                  19   22

%e        |                                  |    |

%e       4|                        16...17...18   23

%e        |                        |              |

%e       3|                        15   26...25...24

%e        |                        |    |

%e       2|              4....5    14   27...28...29

%e        |              |    |    |              |

%e       1|              3    6    13...12...11   30

%e        |              |    |              |    |

%e       0|    0....1....2    7....8....9....10   31..

%e        |

%e     ---+----------------------------------------

%e     y/x|    0    1    2    3    4    5    6    7

%e - hence a(9) = a(12) = a(17) = a(26) = a(27) = 5.

%o (PARI) See Links section.

%Y Cf. A020986, A334577.

%K nonn

%O 0,3

%A _Rémy Sigrist_, May 06 2020