A334221 -id:A334221 - cp's OEIS frontend

A334220 a(n) is the X-coordinate of the point at n steps forward from the origin along the space filling curve U defined in A334188; sequence A334221 gives Y-coordinates.

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

Offset: 0

Rémy Sigrist, Apr 19 2020

			The "forward" branch of curve U starting from the origin is as follows:
     |
    6|                                      25..
     |                                       |
    5|                                      24
     |                                       |
    4|                            21...22...23
     |                             |
    3|                            20   15...14
     |                             |    |    |
    2|                   5....6   19   16   13
     |                   |    |    |    |    |
    1|                   4    7   18...17   12
     |                   |    |              |
    0|    0....1....2....3    8....9...10...11
  ---+----------------------------------------
  y/x|    0    1    2    3    4    5    6    7
- hence a(10) = a(15) = a(16) = a(17) = a(22) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..8066
Rémy Sigrist, PARI program for A334220
Index entries for sequences related to coordinates of 2D curves

Cf. A334188, A334221.

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A334188(A334221(n), a(n)) = n.

A334188 T(n, k) is the number of steps from the point (0, 0) to the point (k, n) along the space filling curve U described in Comments section; a negative value corresponds to moving backwards; square array T(n, k), n, k >= 0 read by antidiagonals downwards.

Original entry on oeis.org

0, 1, -1, 2, -6, -2, 3, -7, -5, -3, 8, 4, -8, -4, -12, 9, 7, 5, -9, -11, -13, 10, 18, 6, -26, -10, -18, -14, 11, 17, 19, -27, -25, -19, -17, -15, 40, 12, 16, 20, -28, -24, -20, -16, -48, 41, 39, 13, 15, 21, -29, -23, -21, -47, -49, 42, 34, 38, 14, 22, -34, -30

Offset: 0

Rémy Sigrist, Apr 18 2020

We start with a unit square U_0 oriented counterclockwise, the origin being at the left bottom corner:
         +---<---+
         |       |
         v       ^
         |       |
         O--->---+
The configuration U_{k+1} is obtained by connecting four copies of the configuration U_k as follows:
             |   |                               |   |
         .   +   +   .                       .   +   +   .
     U_k     ^   v     U_k                       ^   v
         .   +   +   .                       .   +   +   .
             |   |                               |   |
    -+->-+---+   +---+->-+-             -+->-+   +   +   +->-+-
                                -->          v   |   |   ^
    -+-<-+---+   +---+-<-+-             -+-<-+   +-<-+   +-<-+-
             |   |
         .   +   +   .                       .   +->-+   .
     U_k     ^   v     U_k                       ^   v
         .   +   +   .                       .   +   +   .
             |   |                               |   |
For any k >= 0, U_k is a closed curve with length 4^(k+1) and visiting every lattice point (x, y) with 0 <= x, y < 2^(k+1).
The space filling curve U corresponds to the limit of U_k as k tends to infinity, and is a variant of H-order curve.
U visits once every lattice points with nonnegative coordinates and has a single connected component.

			Square array starts:
  n\k|    0    1    2    3    4    5    6    7
  ---+----------------------------------------
    0|    0....1....2....3    8....9...10...11
     |    |              |    |              |
    1|   -1   -6...-7    4    7   18...17   12
     |    |    |    |    |    |    |    |    |
    2|   -2   -5   -8    5....6   19   16   13
     |    |    |    |              |    |    |
    3|   -3...-4   -9  -26..-27   20   15...14
     |              |    |    |    |
    4|  -12..-11..-10  -25  -28   21...22...23
     |    |              |    |              |
    5|  -13  -18..-19  -24  -29  -34..-35   24
     |    |    |    |    |    |    |    |    |
    6|  -14  -17  -20  -23  -30  -33  -36   25..
     |    |    |    |    |    |    |    |
    7|  -15..-16  -21..-22  -31..-32  -37 -102..
     |                                  |    |

Rémy Sigrist, Table of n, a(n) for n = 0..5049
Rémy Sigrist, Representation of U_k for k = 0..5
Rémy Sigrist, Colored representation of U_7
Rémy Sigrist, Colored representation of the table for 0 <= x, y, <= 1023 (where the hue is function of T(y, x))
Rémy Sigrist, PARI program for A334188

See A163334, A323335 and A334232 for similar sequences.

See A334220, A334221, A334222 and A334223 for the coordinates of the curve.

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cp's OEIS Frontend

A334220 a(n) is the X-coordinate of the point at n steps forward from the origin along the space filling curve U defined in A334188; sequence A334221 gives Y-coordinates.

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