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

0,7

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.

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

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.

Example

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

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A026998 A326812 A324893 * A080061 A246595 A209571

Adjacent sequences: A301623 A301624 A301625 * A301627 A301628 A301629

Keyword

nonn,tabl,look

Author

Rémy Sigrist, Mar 24 2018

Status

approved