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A334188 T(n, k) is the number of steps from the point (0, 0) to the point (k, n) along the space filling curve U described in Comments section; a negative value corresponds to moving backwards; square array T(n, k), n, k >= 0 read by antidiagonals downwards. 6
0, 1, -1, 2, -6, -2, 3, -7, -5, -3, 8, 4, -8, -4, -12, 9, 7, 5, -9, -11, -13, 10, 18, 6, -26, -10, -18, -14, 11, 17, 19, -27, -25, -19, -17, -15, 40, 12, 16, 20, -28, -24, -20, -16, -48, 41, 39, 13, 15, 21, -29, -23, -21, -47, -49, 42, 34, 38, 14, 22, -34, -30 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

We start we a unit square U_0 oriented counter-clockwise, the origin being at the left bottom corner:

         +---<---+

         |       |

         v       ^

         |       |

         O--->---+

The configuration U_{k+1} is obtained by connecting four copies of the configuration U_k as follows:

             |   |                               |   |

         .   +   +   .                       .   +   +   .

     U_k     ^   v     U_k                       ^   v

         .   +   +   .                       .   +   +   .

             |   |                               |   |

    -+->-+---+   +---+->-+-             -+->-+   +   +   +->-+-

                                -->          v   |   |   ^

    -+-<-+---+   +---+-<-+-             -+-<-+   +-<-+   +-<-+-

             |   |

         .   +   +   .                       .   +->-+   .

     U_k     ^   v     U_k                       ^   v

         .   +   +   .                       .   +   +   .

             |   |                               |   |

For any k >= 0, U_k is a closed curve with length 4^(k+1) and visiting every lattice point (x, y) with 0 <= x, y < 2^(k+1).

The space filling curve U corresponds to the limit of U_k as k tends to infinity, and is a variant of H-order curve.

U 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

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

Rémy Sigrist, Colored representation of U_7

Rémy Sigrist, Colored representation of the table for 0 <= x, y, <= 1023 (where the hue is function of T(y, x))

Rémy Sigrist, PARI program for A334188

EXAMPLE

Square array starts:

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

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

    0|    0....1....2....3    8....9...10...11

     |    |              |    |              |

    1|   -1   -6...-7    4    7   18...17   12

     |    |    |    |    |    |    |    |    |

    2|   -2   -5   -8    5....6   19   16   13

     |    |    |    |              |    |    |

    3|   -3...-4   -9  -26..-27   20   15...14

     |              |    |    |    |

    4|  -12..-11..-10  -25  -28   21...22...23

     |    |              |    |              |

    5|  -13  -18..-19  -24  -29  -34..-35   24

     |    |    |    |    |    |    |    |    |

    6|  -14  -17  -20  -23  -30  -33  -36   25..

     |    |    |    |    |    |    |    |

    7|  -15..-16  -21..-22  -31..-32  -37 -102..

     |                                  |    |

PROG

(PARI) See Links section.

CROSSREFS

See A163334, A323335 and A334232 for similar sequences.

See A334220, A334221, A334222 and A334223 for the coordinates of the curve.

Sequence in context: A110218 A316259 A057892 * A265993 A115009 A151944

Adjacent sequences:  A334185 A334186 A334187 * A334189 A334190 A334191

KEYWORD

sign,look,tabl

AUTHOR

Rémy Sigrist, Apr 18 2020

STATUS

approved

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Last modified October 1 16:22 EDT 2020. Contains 337443 sequences. (Running on oeis4.)