A341120 a(n) is the X-coordinate of the n-th point of the space filling curve C defined in Comments section; A341121 gives Y-coordinates.

0, 0, 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 2, 2, 3, 4, 4, 3, 3, 3, 4, 5, 5, 5, 4, 4, 5, 6, 6, 6, 7, 8, 8, 7, 7, 7, 8, 8, 7, 6, 6, 5, 5, 5, 6, 5, 5, 5, 6, 6, 7, 8, 8, 9, 9, 9, 8, 8, 9, 10, 10, 10, 11, 12, 12, 11, 11, 11, 12, 12, 13

Offset

0,4

Comments

We define the family {C_k, k >= 0}, as follows:

- C_0 corresponds to the points (0, 0), (0, 1), (1, 1), (2, 1) and (2, 0), in that order:

+---+---+

| |

+ +

O

- for any k >= 0, C_{k+1} is obtained by arranging 4 copies of C_k as follows:

+ . . . + . . . +

. B . B .

+ . . . + . . .

. B . .A C.A C.

. . --> + . . . + . . . +

.A C. .C . A.

+ . . . + . B.B .

O .A . C.

+ . . . + . . . +

O

- the space filling curve C is the limit of C_{2*k} as k tends to infinity.

The even bisection of the curve M defined in A341018 is similar to C and vice versa.

The third quadrisection of C is similar to the Hilbert Hamiltonian walk H = A059252, A059253.

H is the number of points in the middle of each unit square in Hilbert's subdivisions, whereas here points are at the starting corner of each unit square. This start is either the bottom left or top right corner depending on how many 180-degree rotations have been applied. These rotations are digit 3's of n written in base 4, hence the formula below adding A283316.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..16384

F. M. Dekking, Recurrent Sets, Advances in Mathematics, vol. 44, no. 1, 1982. See section 4.8.

David Hilbert, Ueber die stetige Abbildung einer Linie auf ein Flächenstück, Mathematische Annalen, volume 38, number 3, 1891, pages 459-460. Also EUDML (link to GDZ).

Rémy Sigrist, Illustration of C_6

Rémy Sigrist, Illustration of the connection of C with the Hilbert Hamiltonian walk

Rémy Sigrist, Illustration of the connection of C with the space filling curve M defined in A341018

Rémy Sigrist, PARI program for A341120

Index entries for sequences related to coordinates of 2D curves

Formula

a(n) = A341121(n) - A059285(n).

a(n) = A341121(n) iff n belongs to A062880.

a(2*n) = A341018(n).

a(4*n) = 2*A341121(n).

a(16*n) = 4*a(n).

a(n) = A059252(n) + A283316(n+1).

A059253(n) = (a(4*n+2)-1)/2.

Example

Points n and their locations X=a(n), Y=A341121(n) begin as follows. n=7 and n=9 are both at X=3,Y=2, and n=11,n=31 both at X=3,Y=4.
      |       |
    4 | 16---17   12--11,31
      |  |         |    |
    3 | 15---14---13   10
      |                 |
    2 |            8---7,9
      |                 |
    1 |  1----2----3    6
      |  |         |    |
  Y=0 |  0         4----5
      +--------------------
       X=0    1    2    3

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A341121 (Y coordinate), A059285 (projection Y-X), A062880 (n on X=Y diagonal).

Cf. A059252, A059253, A283316, A341018.

Sequence in context: A364200 A059253 A108133 * A332252 A238268 A194883

Adjacent sequences: A341117 A341118 A341119 * A341121 A341122 A341123

Keyword

nonn,changed

Author

Kevin Ryde and Rémy Sigrist, Feb 05 2021

Status

approved