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.

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, -8, -10, -14, 17, 15, 23, 9, -23, -9, -17, -15, 48, 18, 22, 24, -24, -22, -18, -16, -48, 49, 47, 19, 21, 25, -25, -21, -19, -47, -49, 52, 50, 46, 20, 28, 26, -26, -20

Offset: 0

Author

Rémy Sigrist, Apr 19 2020

Keywords

• sign
• tabl

Comments

The H-order curve is built as follows:

- we start we a unit square H_0 oriented counterclockwise, the origin being at the left bottom corner:

+---<---+

| |

v ^

| |

O--->---+

- the configuration H_{k+1} is obtained by connecting four copies of the configuration H_k as follows:

| | | |

. + + . . + + .

H_k ^ v H_k ^ v

. + + . . + + .

| | | |

-+->-+---+ +---+->-+- -+->-+ +-<-+ +->-+-

--> v ^

-+-<-+---+ +---+-<-+- -+-<-+ +->-+ +-<-+-

| | | |

. + + . . + + .

H_k ^ v H_k ^ v

. + + . . + + .

| | | |

- the H-order curve corresponds to the limit of H_k as k tends to infinity,

- the H-order curve visits once every lattice points with nonnegative coordinates and has a single connected component.

Examples

			Square array starts:
  n\k|    0    1    2    3    4    5    6    7
  ---+----------------------------------------
    0|    0....1    4....5   12...13   16...17
     |    |    |    |    |    |    |    |    |
    1|   -1    2....3    6   11   14...15   18
     |    |              |    |              |
    2|   -2   -5...-6    7   10   23...22   19
     |    |    |    |    |    |    |    |    |
    3|   -3...-4   -7    8....9   24   21...20
     |              |              |
    4|  -12..-11   -8  -23..-24   25   28...29
     |    |    |    |    |    |    |    |    |
    5|  -13  -10...-9  -22  -25   26...27   30
     |    |              |    |              |
    6|  -14  -17..-18  -21  -26  -29..-30   31
     |    |    |    |    |    |    |    |    |
    7|  -15..-16  -19..-20  -27..-28  -31   32
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..5049
• GeoWave Developper Guide, Spatial Index
• Rémy Sigrist, Representation of H_k for k = 0..5
• Rémy Sigrist, PARI program for A334232

Crossrefs

See A334188 for a similar sequence.

See A334233, A334234, A334235 and A334236 for the coordinates of the curve.

Programs

PARI

See Links section.