A389948 Shell enumeration of the upper halfplane. Terms are interleaved x and y coordinates.

0, 0, 1, 0, 1, 1, 0, 1, -1, 1, -1, 0, -2, 0, -2, 1, -2, 2, -1, 2, 0, 2, 1, 2, 2, 2, 2, 1, 2, 0, 3, 0, 3, 1, 3, 2, 3, 3, 2, 3, 1, 3, 0, 3, -1, 3, -2, 3, -3, 3, -3, 2, -3, 1, -3, 0, -4, 0, -4, 1, -4, 2, -4, 3, -4, 4, -3, 4, -2, 4, -1, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 4, 3, 4, 2, 4, 1, 4, 0, 5, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5

Offset

1,13

Links

Table of n, a(n) for n=1..101.

Example

Path begins:
      25--24--23--22--21--20--19    Y=3
       |                       |
      26   9--10--11--12--13  18    Y=2
       |   |               |   |
      27   8   5---4---3  14  17    Y=1
       |   |   |       |   |   |
      28   7---6   1---2  15--16    Y=0
     ----------------------------
  X = -3  -2  -1   0   1   2   3
The map starts: (0, 0), (1, 0), (1, 1), (0, 1), (-1, 1), (-1, 0), (-2, 0), (-2, 1), (-2, 2), (-1, 2), (0, 2), (1, 2), (2, 2), (2, 1), (2, 0), (3, 0), (3, 1), (3, 2), (3, 3), (2, 3), (1, 3), (0, 3), (-1, 3), (-2, 3), (-3, 3), (-3, 2), (-3, 1), (-3, 0), (-4, 0), (-4, 1), (-4, 2), (-4, 3), (-4, 4), (-3, 4), (-2, 4), ...
The enumeration can be seen as shells growing around the origin:
  (0, 0);
  (1, 0), (1, 1), (0, 1), (-1, 1), (-1, 0);
  (-2, 0), (-2, 1), (-2, 2), (-1, 2), (0, 2), (1, 2), (2, 2), (2, 1), (2, 0);
  (3, 0), (3, 1), (3, 2), (3, 3), (2, 3), (1, 3), (0, 3), (-1, 3), (-2, 3), (-3, 3), (-3, 2), (-3, 1), (-3, 0);

Prog

(Python)
from itertools import chain
def traverse_upper_halfplane(n):
    points = [(0, 0)]
    arch = 1
    sign = 1
    while len(points) < n:
        for y in range(arch+1):
            points.append((sign*arch, y))
        for x in range(arch-1, -arch-1, -1):
            points.append((sign*x, arch))
        for y in range(arch-1, -1, -1):
            points.append((-sign*arch, y))
        arch += 1
        sign = - sign
    return points[:n]
print(list(chain(*traverse_upper_halfplane(100))))

Crossrefs

Cf. A389649 (x coordinate), A389648 (y coordinate), A389352 (Manhattan distance), A319514 (first quadrant), A296030 (whole plane).

Sequence in context: A305576 A129679 A319514 * A141454 A301564 A334361

Adjacent sequences: A389945 A389946 A389947 * A389949 A389950 A389951

Keyword

sign

Author

Jens Ahlström, Oct 20 2025

Status

approved