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%I #23 Mar 27 2018 18:33:29
%S 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,
%T 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,
%U 10,4,2,4,4,2,4,9,17,25,25,17,9,4,2,4,5,1,1
%N 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).
%C The distance between two Gaussian integers is not necessarily integer, hence the use of the square of the distance.
%C This sequence is a complex variant of A074989.
%C See A301636 for the square array dealing with squares of Gaussian integers.
%H Rémy Sigrist, <a href="/A301626/b301626.txt">Table of n, a(n) for n = 0..20300</a>
%H Rémy Sigrist, <a href="/A301626/a301626.png">Colored scatterplot for abs(x) <= 500 and abs(y) <= 500</a> (where the hue is function of sqrt(T(abs(x), abs(y))))
%H Rémy Sigrist, <a href="/A301626/a301626_1.png">Voronoi diagram of the cubes of Gaussian integers for abs(x) <= 500 and abs(y) <= 500</a>
%H Rémy Sigrist, <a href="/A301626/a301626_2.png">Scatterplot of (x, y) such that T(abs(x), abs(y)) is a square and abs(x) <= 500 and abs(y) <= 500</a>
%H Rémy Sigrist, <a href="/A301626/a301626.gp.txt">PARI program for A301626</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gaussian_integer">Gaussian integer</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Voronoi_diagram">Voronoi diagram</a>
%H <a href="/index/Di#distance_to_the_nearest">Index entries for sequences related to distance to nearest element of some set</a>
%F T(n, k) = T(k, n).
%F T(n, 0) <= A074989(n)^2.
%F T(n, 0) = 0 iff n is a cube (A000578).
%F T(n, k) = 0 iff n + k*i = z^3 for some Gaussian integer z.
%e Square array begins:
%e n\k| 0 1 2 3 4 5 6 7 8 9 10
%e ---+-------------------------------------------------------
%e 0| 0 0 1 4 8 9 4 1 0 1 4 --> A301639
%e 1| 0 1 1 2 5 10 5 2 1 2 2
%e 2| 1 1 0 1 4 9 8 5 4 4 1
%e 3| 4 2 1 2 5 10 13 10 9 5 2
%e 4| 8 5 4 5 8 13 20 17 13 8 5
%e 5| 9 10 9 10 13 18 25 25 18 13 10
%e 6| 4 5 8 13 20 25 32 32 25 20 17
%e 7| 1 2 5 10 17 25 32 41 34 29 26
%e 8| 0 1 4 9 13 18 25 34 45 40 37
%e 9| 1 2 4 5 8 13 20 29 40 53 50
%e 10| 4 2 1 2 5 10 17 26 37 50 65
%o (PARI) See Links section.
%Y Cf. A000578, A074989, A301636, A301639 (first row/column).
%K nonn,tabl,look
%O 0,7
%A _Rémy Sigrist_, Mar 24 2018
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