A301626 -id:A301626 - cp's OEIS frontend

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

Rémy Sigrist, Mar 25 2018

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.

			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

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

Cf. A000290, A001105, A053188, A301626.

PARI
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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.

A301639 a(n) = square of the distance from n to nearest cube of a Gaussian integer.

Original entry on oeis.org

0, 0, 1, 4, 8, 9, 4, 1, 0, 1, 4, 4, 5, 8, 13, 20, 29, 40, 53, 64, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 130, 117, 106, 97, 90, 85, 82, 81, 82, 85, 90, 97, 106, 117, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25

Offset: 0

Rémy Sigrist, Mar 25 2018

nonn

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

This sequence is a variant of A074989: here we minimize norm(n - z^3) where z runs through every Gaussian integers, there we minimize abs(n - m^3) where m runs through every integers.

			For n = 4: the nearest Gaussian cubes to 4 are 2 + 2*i and 2 - 2*i, hence a(4) = (4-2)^2 + 2^2 = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of this sequence versus A074989^2 for n = 0..1000
Wikipedia, Gaussian integer
Index entries for sequences related to distance to nearest element of some set

Cf. A074989, A301626.

a(n) = A301626(n, 0).

a(n) <= A074989(n)^2.

cp's OEIS Frontend

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).

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