%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 Horder 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
