A341120 -id:A341120 - cp's OEIS frontend

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

0, 1, 1, 1, 0, 0, 1, 2, 2, 2, 3, 4, 4, 3, 3, 3, 4, 4, 5, 6, 6, 7, 7, 7, 6, 7, 7, 7, 6, 6, 5, 4, 4, 4, 5, 6, 6, 7, 7, 7, 6, 7, 7, 7, 6, 6, 5, 4, 4, 3, 3, 3, 4, 4, 3, 2, 2, 2, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2, 1, 0, 0, 1, 1, 1, 0, 0, 1

Offset: 0

Kevin Ryde and Rémy Sigrist, Feb 05 2021

nonn

We define the family {C_n, n >= 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 n >= 0, C_{n+1} is obtained by arranging 4 copies of C_n 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*n} as n tends to infinity.

			Points n and their locations X=A341120(n), Y=a(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.
Rémy Sigrist, PARI program for A341121
Index entries for sequences related to coordinates of 2D curves

Cf. A059252, A341019, A341120.

PARI
```
See Links section.
```

a(2*n) = A341019(n).
a(4*n) = 2*A341120(n).
a(16*n) = 4*a(n).
a(n) = A059253(n) + A283316(n+1).
A059252(n) = (a(4*n+2)-1)/2.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula