A059252 -id:A059252 - cp's OEIS frontend

A163900 Squared distance between n's location in A054238 array and A163357 array.

0, 0, 1, 1, 8, 18, 5, 5, 4, 2, 9, 5, 2, 2, 9, 9, 0, 2, 1, 1, 0, 0, 1, 1, 4, 4, 9, 1, 2, 4, 5, 9, 16, 10, 25, 17, 16, 16, 25, 9, 36, 36, 49, 25, 10, 4, 5, 1, 10, 18, 5, 5, 10, 16, 17, 25, 10, 20, 25, 29, 36, 36, 25, 49, 128, 162, 113, 113, 128, 128, 113, 145, 100, 100, 89, 113, 162

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

A. Karttunen, Table of n, a(n) for n = 0..65535

Positions of zeros: A163901. See also A163898, A163899.

a(n) = A000290(abs(A059906(n)-A059252(n))) + A000290(abs(A059905(n)-A059253(n))).

A163539 The change in Y-coordinate when moving from the n-1:th to the n-th term in the type I Hilbert's Hamiltonian walk A163357.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, -1, 0, 1, 0, -1, -1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0, -1, 0, 1, 1, 1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0, -1, 0, 1, 0, 0, -1, 0, -1, -1, 0, 1, 0, -1, 0, 1, 1, 0, 1, 0, 1, 1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0, -1, 0, 1, 1, 0, 1, 0, 1, 1, 0, -1, 0, 1, 0, -1, -1, 0

Offset: 0

Antti Karttunen, Aug 01 2009

sign

A. Karttunen, Table of n, a(n) for n = 0..65536

These are the first differences of A059252. See also: A163538, A163541, A163543.

a(0)=0, a(n) = A059252(n) - A059252(n-1).

A165466 Squared distance between n's location in A163334 array and A163359 array.

Original entry on oeis.org

0, 2, 2, 2, 2, 10, 10, 2, 0, 0, 2, 10, 20, 10, 10, 18, 32, 32, 50, 74, 100, 100, 72, 50, 32, 50, 50, 34, 20, 20, 16, 16, 16, 10, 4, 4, 2, 4, 8, 8, 8, 10, 20, 18, 20, 20, 26, 50, 50, 40, 20, 20, 20, 20, 32, 32, 34, 40, 58, 74, 100, 74, 74, 80, 80, 80, 52, 52, 50, 34, 34, 32

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Equivalently, squared distance between n's location in A163336 array and A163357 array. See example at A166043.

A. Karttunen, Table of n, a(n) for n = 0..65535

Positions of zeros: A165467. See also A166043, A165464, A163897, A163900.

a(n) = A000290(abs(A163529(n)-A059253(n))) + A000290(abs(A163528(n)-A059252(n))).

A165464 Squared distance between n's location in A163334 array and A163357 array.

Original entry on oeis.org

0, 0, 2, 4, 2, 4, 2, 0, 0, 2, 2, 4, 4, 4, 2, 0, 0, 2, 2, 4, 4, 2, 0, 0, 0, 0, 2, 4, 4, 2, 8, 10, 16, 16, 4, 2, 2, 10, 16, 10, 8, 8, 20, 20, 20, 18, 18, 32, 18, 10, 4, 2, 4, 10, 8, 2, 2, 10, 10, 4, 4, 4, 2, 10, 16, 26, 20, 10, 2, 4, 10, 18, 32, 32, 50, 52, 52, 34, 40, 58, 80, 80, 106, 146

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Equivalently, squared distance between n's location in A163336 array and A163359 array. See example at A166041.

A. Karttunen, Table of n, a(n) for n = 0..65535

Positions of zeros: A165465. See also A165466, A163897, A163900.

a(n) = A000290(abs(A163529(n)-A059252(n))) + A000290(abs(A163528(n)-A059253(n))).

A275103 Hilbert curve constructed by greedy algorithm, such that each element is the smallest positive integer possible and that all rows, columns, and diagonals contain distinct numbers.

Original entry on oeis.org

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

Offset: 0

Kerry Mitchell, Jul 16 2016

nonn

The n-th cell has x-coordinates given by A059252 and y-coordinates given by A059253.

This idea is similar to A269526 and A274640, but for a different curve.

			The Hilbert curve begins:
  1,   4,   2,   3, ...
  2,   3,   5,   1, ...
  5,   6,   4,   2, ...
  4,   2,   1,   5, ...
...

Kerry Mitchell, Table of n, a(n) for n = 0..4095

Cf. A269526 uses antidiagonals instead of the Hilbert curve and A274640 uses a square spiral.

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.

Original entry on oeis.org

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
```
See Links section.
```

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
```
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.

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.

A385414 Number of distinct states of Conway's Game of Life, starting from an n-th level Hilbert curve on a toroidal 2^(n+1)-1 by 2^(n+1)-1 grid.

Original entry on oeis.org

2, 2, 3, 24, 70, 584, 1325, 2082, 5304, 6327, 10679, 11822

Offset: 0

Luke Bennet, Jun 27 2025

The curve is taken with segments of length 2 so that it follows a path through coordinates (2*A059252(t), 2*A059253(t)) for 0 <= t < 2^n.
The size of the toroidal grid is the extent of these coordinates so that the cells on one edge are immediately adjacent to the cells on the opposite side.
The grid has a fixed position and orientation and states differing at any cell are distinct.

			For n=0, the curve is a single cell on a 1 X 1 toroidal grid and has a(0) = 2 states: initially live, then dead and remaining so.
For n=2 the initial state and two subsequent states are
  o o o . o o o | . . . . . . . | . . . . . . . |
  o . o . o . o | . . . . . . . | . . . . . . . |
  o . o o o . o | . . o . o . . | . . . . . . . |
  o . . . . . o | . . . . . . . | . . . . . . . |
  o o o . o o o | . . o . o . . | . . . . . . . |
  . . o . o . . | . . . . . . . | . . . . . . . |
  o o o . o o o | . . . . . . . | . . . . . . . |
  (generation 1)  (generation 2)  (generation 3)
Every generation after 3 is identical to generation 3, so this sequence has 3 distinct states. Thus, a(2) = 3.

Cf. A059252, A059253.

cp's OEIS Frontend

A163900 Squared distance between n's location in A054238 array and A163357 array.

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A163539 The change in Y-coordinate when moving from the n-1:th to the n-th term in the type I Hilbert's Hamiltonian walk A163357.

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A165466 Squared distance between n's location in A163334 array and A163359 array.

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Formula

A165464 Squared distance between n's location in A163334 array and A163357 array.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A275103 Hilbert curve constructed by greedy algorithm, such that each element is the smallest positive integer possible and that all rows, columns, and diagonals contain distinct numbers.

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Author

Keywords

Comments

Examples

Links

Crossrefs

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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Comments

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PARI

Formula

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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PARI

Formula

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.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A385414 Number of distinct states of Conway's Game of Life, starting from an n-th level Hilbert curve on a toroidal 2^(n+1)-1 by 2^(n+1)-1 grid.

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