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