A059285 -id:A059285 - cp's OEIS frontend

A059252 Hilbert's Hamiltonian walk on N X N projected onto x axis: m(3).

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

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 23 2001

nonn

This is the X-coordinate of the n-th term in Hilbert's Hamiltonian walk A163359 and the Y-coordinate of its transpose A163357.

			[m(1)=0 0 1 1, m'(1)= 0 1 10] [m(2) =0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, m'(2)=0 1 1 0 0 0 1 1 2 2 3 3 3 2 2 3].

Antti Karttunen, Table of n, a(n) for n = 0..65535
Joerg Arndt, Matters Computational (The Fxtbook), section 1.31.1 "The Hilbert curve", page 85, lin2hilbert.
Michael Beeler, R. William Gosper, and Richard Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory report AIM-239, February 1972. Item 115 by Gosper, page 52. Also HTML transcription. (To use algorithm S or the state table, pad n with high 0-bits to a multiple of 4 bits.)
J. Shallit, Hilbert's spacefilling curve described by automatic, regular, and synchronized sequences, arXiv:2106.01062 [cs.FL], June 2 2021.
Index entries for sequences related to coordinates of 2D curves

See also the y-projection, m'(3), A059253, as well as: A163539, A163540, A163542, A059261, A059285, A163547 and A163529.

void h(unsigned int *x, unsigned int *y, unsigned int l){
x[0] = y[0] = 0; unsigned int *t = NULL; unsigned int n = 0, k = 0;
for(unsigned int i = 1; i>(2*n)){
case 1: x[i] = y[i&k]; y[i] = x[i&k]+(1<Jared Rager, Jan 09 2021 */
(C++) See Fxtbook link.

Initially [m(0) = 0, m'(0) = 0]; recursion: m(2n + 1) = m(2n).m'(2n).f(m'(2n), 2n).c(m(2n), 2n + 1); m'(2n + 1) = m'(2n).f(m(2n), 2n).f(m(2n), 2n).mir(m'(2n)); m(2n) = m(2n - 1).f(m'(2n - 1), 2n - 1).f(m'(2n - 1), 2n - 1).mir(m(2n - 1)); m'(2n) = m'(2n - 1).m(2n - 1).f(m(2n - 1), 2n - 1).c(m'(2n - 1), 2n); where f(m, n) is the alphabetic morphism i := i + 2^n [example: f(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 2) = 4 4 5 5 6 7 7 6 6 7 7 6 5 5 4 4]; c(m, n) is the complementation to 2^n - 1 alphabetic morphism [example: c(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 3) = 7 7 6 6 5 4 4 5 5 4 4 5 6 6 7 7]; and mir(m) is the mirror operator [example: mir(0 1 1 0 0 0 1 1 2 2 3 3 3 2 2 3) = 3 2 2 3 3 3 2 2 1 1 0 0 0 1 1 0].

a(n) = A002262(A163358(n)) = A025581(A163360(n)) = A059906(A163356(n)).

Extended by Antti Karttunen, Aug 01 2009

A059253 Hilbert's Hamiltonian walk on N X N projected onto y axis: m'(3).

Original entry on oeis.org

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

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 23 2001

nonn

This is the Y-coordinate of the n-th term in the type I Hilbert's Hamiltonian walk A163359 and the X-coordinate of its transpose A163357.

Antti Karttunen, Table of n, a(n) for n = 0..65535
J. Shallit, Hilbert's spacefilling curve described by automatic, regular, and synchronized sequences, arXiv:2106.01062 [cs.FL], June 2 2021.
Index entries for sequences related to coordinates of 2D curves

See also the y-projection, m(3), A059252 as well as A163538, A163540, A163542, A059261, A059285, A163547 and A163528.

C
```
See A059252.
```

a(n) = A025581(A163358(n)) = A002262(A163360(n)) = A059905(A163356(n)).

Extended by Antti Karttunen, Aug 01 2009

A059261 Hilbert's Hamiltonian walk on N X N projected onto the first diagonal: M(3) (sum of the sequences A059252 and A059253).

Original entry on oeis.org

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

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 24 2001

nonn

The interest comes from a simplest recursion than the cross-recursion, dependent on parity, governing the projections onto the x and y axis.

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

Cf. the x-projection m(3), A059252 and the y-projection m'(3), A059253. See also: A163530, A059285, A163547.

Initially, M(0)=0; recursion: M(n+1)=M(n).f(M(n), n).f(M(n), n+1).d(M(n), n); -f(m, n) is the alphabetic morphism i := i+2^n; [example: f(0 1 2 1 2 3 4 3 4 5 6 5 4 3 2 3, 2)=4 5 6 5 6 7 8 7 8 9 10 9 8 7 6 7 ] -d(m, n) is the complementation to 2^(n-1)*3-2, alphabetic morphism; [example: d(0 1 2 1 2 3 4 3 4 5 6 5 4 3 2 3, 3)=10 9 8 9 8 7 6 7 6 5 4 5 6 7 8 7] Here is M(3). [M(1)=0.1.2.1, M(2)=0 1 2 1.2 3 4 3.4 5 6 5.4 3 2 3]

Extended by Antti Karttunen, Aug 01 2009

A163530 a(n) = A163528(n)+A163529(n).

Original entry on oeis.org

0, 1, 2, 3, 2, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 4, 5, 6, 7, 8, 9, 8, 7, 8, 9, 10, 11, 10, 9, 10, 11, 12, 13, 12, 11, 10, 9, 8, 7, 8, 9, 8, 7, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 6, 7, 8, 9, 8, 7, 8, 9, 10, 11, 12, 13, 12, 11, 10, 9, 10, 11, 12, 13, 14, 15, 14, 13, 14, 15, 16, 17, 18, 19, 18, 17

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

A. Karttunen, Table of n, a(n) for n = 0..59048

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.

cp's OEIS Frontend

A059252 Hilbert's Hamiltonian walk on N X N projected onto x axis: m(3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Formula

Extensions

A059253 Hilbert's Hamiltonian walk on N X N projected onto y axis: m'(3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

C

Formula

Extensions

A059261 Hilbert's Hamiltonian walk on N X N projected onto the first diagonal: M(3) (sum of the sequences A059252 and A059253).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A163530 a(n) = A163528(n)+A163529(n).

Views

Author

Keywords

Links

Crossrefs

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