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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). 2
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 (list; table; graph; refs; listen; history; text; internal format)
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: A267486 A285229 A227425 * A238857 A253587 A110962

Adjacent sequences:  A301633 A301634 A301635 * A301637 A301638 A301639

KEYWORD

nonn,tabl

AUTHOR

Rémy Sigrist, Mar 25 2018

STATUS

approved

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Last modified January 28 13:24 EST 2020. Contains 331321 sequences. (Running on oeis4.)