A334188 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.

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, -10, -18, -14, 11, 17, 19, -27, -25, -19, -17, -15, 40, 12, 16, 20, -28, -24, -20, -16, -48, 41, 39, 13, 15, 21, -29, -23, -21, -47, -49, 42, 34, 38, 14, 22, -34, -30

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

See A163334, A323335 and A334232 for similar sequences.

See A334220, A334221, A334222 and A334223 for the coordinates of the curve.

Sequence in context: A110218 A316259 A057892 * A265993 A115009 A151944

Adjacent sequences: A334185 A334186 A334187 * A334189 A334190 A334191

Keyword

sign,look,tabl

Author

Rémy Sigrist, Apr 18 2020

Status

approved