A301636 Square array T(n, k) read by antidiagonals upwards, n >= 0 and k >= 0: T(n, k) = square of the distance from n + k*i to nearest square of a Gaussian integer (where i denotes the root of -1 with positive imaginary part).

0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 2, 4, 2, 4, 1, 1, 4, 2, 4, 9, 4, 2, 4, 1, 1, 5, 4, 4, 5, 5, 2, 0, 2, 5, 1, 1, 5, 8, 5, 1, 1, 5, 2, 0, 0, 2, 8, 10, 4, 2, 4, 5, 1, 1, 1, 1, 5, 10, 8, 5, 5, 9, 4, 2, 4, 4, 2, 4, 9, 5, 5, 8, 10, 9, 5, 5, 9, 9, 5, 5, 9, 4, 2, 4, 10

Offset

0,8

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 A053188.

See A301626 for the square array dealing with cubes of Gaussian integers.

Links

Table of n, a(n) for n=0..85.

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 squares 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 A301636

Wikipedia, Gaussian integer

Wikipedia, Voronoi diagram

Index entries for sequences related to distance to nearest element of some set

Formula

T(n, 0) <= A053188(n)^2.

T(n, 0) = 0 iff n is a square (A000290).

T(0, k) = 0 iff k is twice a square (A001105).

T(n, k) = 0 iff n + k*i = z^2 for some Gaussian integer z.

Example

Square array begins:
  n\k|    0    1    2    3    4    5    6    7    8    9   10
  ---+-------------------------------------------------------
    0|    0    1    0    1    4    9    4    1    0    1    4
    1|    0    1    1    2    4    5    5    2    1    2    5
    2|    1    2    4    2    1    2    5    5    4    5    8
    3|    1    2    4    1    0    1    4    9    9   10    8
    4|    0    1    4    2    1    2    5   10   16   10    5
    5|    1    2    5    5    4    5    8   10   13    9    4
    6|    4    5    8   10    8    5    4    5    8   10    5
    7|    4    5    8   10    5    2    1    2    5   10    8
    8|    1    2    5    9    4    1    0    1    4    9   13
    9|    0    1    4    9    5    2    1    2    5   10   17
   10|    1    2    5   10    8    5    4    5    8   13   20

Prog

(PARI) See Links section.

Crossrefs

Cf. A000290, A001105, A053188, A301626.

Sequence in context: A285229 A227425 A333213 * A238857 A253587 A110962

Adjacent sequences: A301633 A301634 A301635 * A301637 A301638 A301639

Keyword

nonn,tabl

Author

Rémy Sigrist, Mar 25 2018

Status

approved