A393021 List points (x,y,z) having integer coordinates, sorted first by L = |x| + |y| + |z| and in case of ties, then by z and last by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives x-coordinates.

0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 1, 2, 1, 0, -1, -2, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 1, 2, 1, 0, -1, -2, -1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -2, -1, 0, 1, 2, 1, 0, -1, -2, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 1, 2, 1, 0, -1, -2, -1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -2, -1, 0

Offset

0,15

Comments

Might be called octahedral spiral.

Links

Hugo Pfoertner, Table of n, a(n) for n = 0..11520

Hugo Pfoertner, Illustration of initial terms.

Example

       a(n)  A393022 A393023
   n    x       y       z    L   phi(degrees)
   0    0       0       0    0     0
   1    0       0      -1    1     0
   2    0       1       0    1     0
   3    1       0       0    1    90
   4    0      -1       0    1   180
   5   -1       0       0    1   270
   6    0       0       1    1     0
   7    0       0      -2    2     0
   8    0       1      -1    2     0
   9    1       0      -1    2    90
  10    0      -1      -1    2   180
  11   -1       0      -1    2   270
  12    0       2       0    2     0
  13    1       1       0    2    45
  14    2       0       0    2    90
  15    1      -1       0    2   135
  16    0      -2       0    2   180
  17   -1      -1       0    2   225
  18   -2       0       0    2   270
  19   -1       1       0    2   315
  20    0       1       1    2     0
  21    1       0       1    2    90
  22    0      -1       1    2   180
  23   -1       0       1    2   270
  24    0       0       2    2     0

Crossrefs

Cf. A001845, A005899, A393022 (y-coordinates), A393023 (z-coordinates).

Cf. A343640, A343641, A343642, A343643 (similar, but using sup norm max(|x|, |y|, |z|) instead of Manhattan distance L).

Sequence in context: A090787 A351559 A229707 * A262680 A366128 A191329

Adjacent sequences: A393018 A393019 A393020 * A393022 A393023 A393024

Keyword

sign

Author

Hugo Pfoertner, Jan 31 2026

Status

approved