A341019 -id:A341019 - cp's OEIS frontend

A341018 a(n) is the X-coordinate of the n-th point of the space filling curve M defined in Comments section; A341019 gives Y-coordinates.

0, 1, 2, 3, 2, 3, 2, 1, 0, 1, 0, 1, 2, 3, 4, 3, 4, 5, 4, 5, 6, 7, 8, 7, 8, 7, 6, 5, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 14, 15, 14, 13, 12, 13, 14, 15, 14, 15, 14, 13, 12, 11, 12, 11, 10, 9, 8, 9, 8, 9, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15

Offset: 0

Rémy Sigrist, Feb 02 2021

nonn

We define the family {M_n, n >= 0}, as follows:
- M_0 corresponds to the points (0, 0), (1, 1) and (2, 0), in that order:
          +
         / \
        /   \
       +     +
      O
- for any n >= 0, M_{n+1} is obtained by arranging 4 copies of M_n as follows:
                           + . . . + . . . +
                           .   B   .   B   .
       + . . . +           .       .       .
       .   B   .           .A     C.A     C.
       .       .    -->    + . . . + . . . +
       .A     C.           .C      .      A.
       + . . . +           .      B.B      .
      O                    .A      .      C.
                           + . . . + . . . +
                          O
- for any n >= 0, M_n has A087289(n) points,
- the space filling curve M is the limit of M_{2*n} as n tends to infinity.
The odd bisection of M is similar to a Hilbert's Hamiltonian walk (hence the connection with A059253, see illustration in Links section).

			The curve M starts as follows:
       11+ 13+   +19 +21
        / \ / \ / \ / \
     10+ 12+ 14+18 +20 +22
        \     / \     /
        9+ 15+   +17 +23
        /     \ /     \
      8+  6+   +   +26 +24
        \ / \ 16  / \ /
        7+  5+   +27 +25
            /     \
          4+       +28
            \     /
        1+  3+   +29 +31
        / \ /     \ / \
      0+  2+       +30 +32
- so a(0) = a(8) = a(10) = 0,
     a(1) = a(7) = a(9) = a(11) = 1.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
F. M. Dekking, Recurrent Sets, Advances in Mathematics, vol. 44, no. 1, 1982.
Larry Riddle, Space Filling Curve
Rémy Sigrist, Illustration of M_6
Rémy Sigrist, Illustration of the connection between the space filling curve M and Hilbert Hamiltonian walk
Rémy Sigrist, PARI program for A341018
Index entries for sequences related to coordinates of 2D curves

Cf. A000695, A059253, A087289, A341019.

PARI
```
See Links section.
```

a(n) = A341019(n) iff n belongs to A000695.
a(2*n-1) + A341019(2*n-1) = a(2*n) + A341019(2*n) for any n > 0.
a(2*n) - A341019(2*n) = a(2*n+1) - A341019(2*n+1) for any n >= 0.
A059253(n) = (a(2*n+1) - 1)/2.
a(4*n) = 2*A341019(n).
a(16*n) = 4*a(n).

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.

Original entry on oeis.org

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

A341018 a(n) is the X-coordinate of the n-th point of the space filling curve M defined in Comments section; A341019 gives Y-coordinates.

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