OFFSET
0,4
COMMENTS
We start with a unit square U_0 oriented counterclockwise, 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
AUTHOR
Rémy Sigrist, Apr 18 2020
STATUS
approved