A341121 -id:A341121 - cp's OEIS frontend

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

Kevin Ryde and Rémy Sigrist, Feb 05 2021

nonn

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.

			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

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

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

Cf, A059252, A341018.

PARI
```
See Links section.
```

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.

cp's OEIS Frontend

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.

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