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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

%I

%S 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,

%T -10,-18,-14,11,17,19,-27,-25,-19,-17,-15,40,12,16,20,-28,-24,-20,-16,

%U -48,41,39,13,15,21,-29,-23,-21,-47,-49,42,34,38,14,22,-34,-30

%N 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.

%C We start with a unit square U_0 oriented counterclockwise, the origin being at the left bottom corner:

%C +---<---+

%C | |

%C v ^

%C | |

%C O--->---+

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

%C | | | |

%C . + + . . + + .

%C U_k ^ v U_k ^ v

%C . + + . . + + .

%C | | | |

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

%C --> v | | ^

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

%C | |

%C . + + . . +->-+ .

%C U_k ^ v U_k ^ v

%C . + + . . + + .

%C | | | |

%C 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).

%C 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.

%C U visits once every lattice points with nonnegative coordinates and has a single connected component.

%H Rémy Sigrist, <a href="/A334188/b334188.txt">Table of n, a(n) for n = 0..5049</a>

%H Rémy Sigrist, <a href="/A334188/a334188.png">Representation of U_k for k = 0..5</a>

%H Rémy Sigrist, <a href="/A334188/a334188_1.png">Colored representation of U_7</a>

%H Rémy Sigrist, <a href="/A334188/a334188_2.png">Colored representation of the table for 0 <= x, y, <= 1023</a> (where the hue is function of T(y, x))

%H Rémy Sigrist, <a href="/A334188/a334188.gp.txt">PARI program for A334188</a>

%e Square array starts:

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

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

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

%e | | | | |

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

%e | | | | | | | | |

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

%e | | | | | | |

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

%e | | | | |

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

%e | | | | |

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

%e | | | | | | | | |

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

%e | | | | | | | |

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

%e | | |

%o (PARI) See Links section.

%Y See A163334, A323335 and A334232 for similar sequences.

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

%K sign,look,tabl

%O 0,4

%A _Rémy Sigrist_, Apr 18 2020

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Last modified April 17 13:31 EDT 2021. Contains 343063 sequences. (Running on oeis4.)