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A334232 T(n, k) is the number of steps from the point (0, 0) to the point (k, n) along the H-order curve; a negative value corresponds to moving backwards; square array T(n, k), n, k >= 0 read by antidiagonals downwards. 6
0, 1, -1, 4, 2, -2, 5, 3, -5, -3, 12, 6, -6, -4, -12, 13, 11, 7, -7, -11, -13, 16, 14, 10, 8, -8, -10, -14, 17, 15, 23, 9, -23, -9, -17, -15, 48, 18, 22, 24, -24, -22, -18, -16, -48, 49, 47, 19, 21, 25, -25, -21, -19, -47, -49, 52, 50, 46, 20, 28, 26, -26, -20 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The H-order curve is built as follows:

- we start we a unit square H_0 oriented counterclockwise, the origin being at the left bottom corner:

         +---<---+

         |       |

         v       ^

         |       |

         O--->---+

- the configuration H_{k+1} is obtained by connecting four copies of the configuration H_k as follows:

             |   |                               |   |

         .   +   +   .                       .   +   +   .

     H_k     ^   v     H_k                       ^   v

         .   +   +   .                       .   +   +   .

             |   |                               |   |

    -+->-+---+   +---+->-+-             -+->-+   +-<-+   +->-+-

                                -->          v           ^

    -+-<-+---+   +---+-<-+-             -+-<-+   +->-+   +-<-+-

             |   |                               |   |

         .   +   +   .                       .   +   +   .

     H_k     ^   v     H_k                       ^   v

         .   +   +   .                       .   +   +   .

             |   |                               |   |

- the H-order curve corresponds to the limit of H_k as k tends to infinity,

- the H-order curve visits once every lattice points with nonnegative coordinates and has a single connected component.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..5049

GeoWave Developper Guide, Spatial Index

Rémy Sigrist, Representation of H_k for k = 0..5

Rémy Sigrist, PARI program for A334232

EXAMPLE

Square array starts:

  n\k|    0    1    2    3    4    5    6    7

  ---+----------------------------------------

    0|    0....1    4....5   12...13   16...17

     |    |    |    |    |    |    |    |    |

    1|   -1    2....3    6   11   14...15   18

     |    |              |    |              |

    2|   -2   -5...-6    7   10   23...22   19

     |    |    |    |    |    |    |    |    |

    3|   -3...-4   -7    8....9   24   21...20

     |              |              |

    4|  -12..-11   -8  -23..-24   25   28...29

     |    |    |    |    |    |    |    |    |

    5|  -13  -10...-9  -22  -25   26...27   30

     |    |              |    |              |

    6|  -14  -17..-18  -21  -26  -29..-30   31

     |    |    |    |    |    |    |    |    |

    7|  -15..-16  -19..-20  -27..-28  -31   32

PROG

(PARI) See Links section.

CROSSREFS

See A334188 for a similar sequence.

See A334233, A334234, A334235 and A334236 for the coordinates of the curve.

Sequence in context: A245518 A217462 A016510 * A244681 A023634 A199609

Adjacent sequences:  A334229 A334230 A334231 * A334233 A334234 A334235

KEYWORD

sign

AUTHOR

Rémy Sigrist, Apr 19 2020

STATUS

approved

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Last modified April 11 09:54 EDT 2021. Contains 342886 sequences. (Running on oeis4.)