A301626 Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0: T(n, k) = square of the distance from n + k*i to nearest cube of a Gaussian integer (where i denotes the root of -1 with positive imaginary part).

0, 0, 0, 1, 1, 1, 4, 1, 1, 4, 8, 2, 0, 2, 8, 9, 5, 1, 1, 5, 9, 4, 10, 4, 2, 4, 10, 4, 1, 5, 9, 5, 5, 9, 5, 1, 0, 2, 8, 10, 8, 10, 8, 2, 0, 1, 1, 5, 13, 13, 13, 13, 5, 1, 1, 4, 2, 4, 10, 20, 18, 20, 10, 4, 2, 4, 4, 2, 4, 9, 17, 25, 25, 17, 9, 4, 2, 4, 5, 1, 1

Offset: 0

Author

Rémy Sigrist, Mar 24 2018

Keywords

• nonn
• tabl
• look

Comments

The distance between two Gaussian integers is not necessarily integer, hence the use of the square of the distance.

This sequence is a complex variant of A074989.

See A301636 for the square array dealing with squares of Gaussian integers.

Examples

			Square array begins:
  n\k|    0    1    2    3    4    5    6    7    8    9   10
  ---+-------------------------------------------------------
    0|    0    0    1    4    8    9    4    1    0    1    4  -->  A301639
    1|    0    1    1    2    5   10    5    2    1    2    2
    2|    1    1    0    1    4    9    8    5    4    4    1
    3|    4    2    1    2    5   10   13   10    9    5    2
    4|    8    5    4    5    8   13   20   17   13    8    5
    5|    9   10    9   10   13   18   25   25   18   13   10
    6|    4    5    8   13   20   25   32   32   25   20   17
    7|    1    2    5   10   17   25   32   41   34   29   26
    8|    0    1    4    9   13   18   25   34   45   40   37
    9|    1    2    4    5    8   13   20   29   40   53   50
   10|    4    2    1    2    5   10   17   26   37   50   65
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..20300
• Rémy Sigrist, Colored scatterplot for abs(x) <= 500 and abs(y) <= 500 (where the hue is function of sqrt(T(abs(x), abs(y))))
• Rémy Sigrist, Voronoi diagram of the cubes of Gaussian integers for abs(x) <= 500 and abs(y) <= 500
• Rémy Sigrist, Scatterplot of (x, y) such that T(abs(x), abs(y)) is a square and abs(x) <= 500 and abs(y) <= 500
• Rémy Sigrist, PARI program for A301626
• Wikipedia, Gaussian integer
• Wikipedia, Voronoi diagram
• Index entries for sequences related to distance to nearest element of some set

Crossrefs

Cf. A000578, A074989, A301636, A301639 (first row/column).

Programs

PARI

See Links section.

Formula

T(n, k) = T(k, n).

T(n, 0) <= A074989(n)^2.

T(n, 0) = 0 iff n is a cube (A000578).

T(n, k) = 0 iff n + k*i = z^3 for some Gaussian integer z.