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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.

%I A334232 #21 Jan 05 2024 12:56:41
%S A334232 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,
%T A334232 -8,-10,-14,17,15,23,9,-23,-9,-17,-15,48,18,22,24,-24,-22,-18,-16,-48,
%U A334232 49,47,19,21,25,-25,-21,-19,-47,-49,52,50,46,20,28,26,-26,-20
%N 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.
%C A334232 The H-order curve is built as follows:
%C A334232 - we start we a unit square H_0 oriented counterclockwise, the origin being at the left bottom corner:
%C A334232          +---<---+
%C A334232          |       |
%C A334232          v       ^
%C A334232          |       |
%C A334232          O--->---+
%C A334232 - the configuration H_{k+1} is obtained by connecting four copies of the configuration H_k as follows:
%C A334232              |   |                               |   |
%C A334232          .   +   +   .                       .   +   +   .
%C A334232      H_k     ^   v     H_k                       ^   v
%C A334232          .   +   +   .                       .   +   +   .
%C A334232              |   |                               |   |
%C A334232     -+->-+---+   +---+->-+-             -+->-+   +-<-+   +->-+-
%C A334232                                 -->          v           ^
%C A334232     -+-<-+---+   +---+-<-+-             -+-<-+   +->-+   +-<-+-
%C A334232              |   |                               |   |
%C A334232          .   +   +   .                       .   +   +   .
%C A334232      H_k     ^   v     H_k                       ^   v
%C A334232          .   +   +   .                       .   +   +   .
%C A334232              |   |                               |   |
%C A334232 - the H-order curve corresponds to the limit of H_k as k tends to infinity,
%C A334232 - the H-order curve visits once every lattice points with nonnegative coordinates and has a single connected component.
%H A334232 Rémy Sigrist, <a href="/A334232/b334232.txt">Table of n, a(n) for n = 0..5049</a>
%H A334232 GeoWave Developper Guide, <a href="http://locationtech.github.io/geowave/devguide.html#spatial-index">Spatial Index</a>
%H A334232 Rémy Sigrist, <a href="/A334232/a334232.png">Representation of H_k for k = 0..5</a>
%H A334232 Rémy Sigrist, <a href="/A334232/a334232.gp.txt">PARI program for A334232</a>
%e A334232 Square array starts:
%e A334232   n\k|    0    1    2    3    4    5    6    7
%e A334232   ---+----------------------------------------
%e A334232     0|    0....1    4....5   12...13   16...17
%e A334232      |    |    |    |    |    |    |    |    |
%e A334232     1|   -1    2....3    6   11   14...15   18
%e A334232      |    |              |    |              |
%e A334232     2|   -2   -5...-6    7   10   23...22   19
%e A334232      |    |    |    |    |    |    |    |    |
%e A334232     3|   -3...-4   -7    8....9   24   21...20
%e A334232      |              |              |
%e A334232     4|  -12..-11   -8  -23..-24   25   28...29
%e A334232      |    |    |    |    |    |    |    |    |
%e A334232     5|  -13  -10...-9  -22  -25   26...27   30
%e A334232      |    |              |    |              |
%e A334232     6|  -14  -17..-18  -21  -26  -29..-30   31
%e A334232      |    |    |    |    |    |    |    |    |
%e A334232     7|  -15..-16  -19..-20  -27..-28  -31   32
%o A334232 (PARI) See Links section.
%Y A334232 See A334188 for a similar sequence.
%Y A334232 See A334233, A334234, A334235 and A334236 for the coordinates of the curve.
%K A334232 sign,tabl
%O A334232 0,4
%A A334232 _Rémy Sigrist_, Apr 19 2020