A341018 -id:A341018 - cp's OEIS frontend

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

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

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 A059252).

			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(2) = a(30) = a(32) = 0,
     a(1) = a(3) = a(29) = a(31) = 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, PARI program for A341019
Index entries for sequences related to coordinates of 2D curves

Cf. A059252, A087289, A341018.

PARI
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A059252(n) = (a(2*n+1)-1)/2.
a(4*n) = 2*A341018(n).
a(16*n) = 4*a(n).

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.

Original entry on oeis.org

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
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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.

A341163 a(n) is the X-coordinate of the n-th point of the space filling curve A defined in Comments section; A341164 gives Y-coordinates.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Feb 06 2021

Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:
           Y
          /
         /
        0 ---- X
We define the family {A_n, n >= 0} as follows:
- A_0 corresponds to the points (0, 0), (1, 1) and (3, 0), in that order:
           .   +__   .
         ----     ----
        +     .     .     +
       0
- for any n >= 0, A_{n+1} is obtained by arranging 4 copies of A_n as follows:
                         +
                        /B\
        +              /   \
       /B\            /A   C\
      /   \    -->   +-------+
     /A   C\        /B\C   B/A\
    +-------+      /   \   /   \
   O              /A   C\A/B   C\
                 +-------+-------+
                O
- the space filling curve A is the limit of A_n as n tends to infinity.
This sequence has similarities with A341018.

			The curve A starts as follows:
                  .
                .   .
              .   5   .
            4   .   .   6
          .   .   3   .   .
        .   1   .   .   7   .
      0   .   .   2   .   .   8
- so a(0) = a(4) = 0,
     a(1) = a(5) = 1,
     a(3) = 2,
     a(2) = a(6) = 3,
     a(8) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Zbigniew Fiedorowicz, The Peano Curve Theorem
Rémy Sigrist, Illustration of A_5
Rémy Sigrist, Illustration of the first bisection of A
Rémy Sigrist, Illustration of the first quadrisection of A
Rémy Sigrist, Illustration of the fourth quadrisection of A
Rémy Sigrist, PARI program for A341163
Index entries for sequences related to coordinates of 2D curves

Cf. A341018, A341164.

PARI
```
See Links section.
```

cp's OEIS Frontend

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

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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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A341163 a(n) is the X-coordinate of the n-th point of the space filling curve A defined in Comments section; A341164 gives Y-coordinates.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI