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%I #9 Apr 24 2023 01:31:49
%S -1,0,1,0,0,1,0,0,0,1,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,2,1,1,0,0,0,0,0,1,
%T 1,1,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,1,1,1,1,1,0,0,
%U 0,0,0,0,0,2,1,1,1,1,0,0,0,0,0,0,0,0,2,1,1,1,1
%N Square array A(n, k), n, k >= 0, read by upwards antidiagonals; if Newton's method applied to the complex function f(z) = z^3 - 1 and starting from n + k*i reaches or converges to exp(2*r*i*Pi/3) for some r in 0..2, then A(n, k) = r, otherwise A(n, k) = -1 (where i denotes the imaginary unit).
%C This sequence is related to the Newton fractal, and exhibits similar rich patterns (see illustration in Links section).
%H Rémy Sigrist, <a href="/A362496/a362496.png">Colored representation of the square array for n, k <= 1000</a> (black, white, blue and red pixels denote, respectively, -1, 0, 1 and 2)
%H Rémy Sigrist, <a href="/A362496/a362496.gp.txt">PARI program</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Newton_fractal">Newton fractal</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Newton%27s_method">Newton's method</a>
%e Array A(n, k) begins:
%e n\k | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e ----+------------------------------------------------------
%e 0 | -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e 1 | 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1
%e 2 | 0 0 0 0 2 1 1 1 1 1 1 1 1 1 1 1
%e 3 | 0 0 0 0 0 2 2 1 1 1 1 1 1 1 1 1
%e 4 | 0 0 0 0 0 0 1 2 2 1 2 1 1 1 1 1
%e 5 | 0 0 0 0 0 0 0 0 2 2 0 1 1 1 1 1
%e 6 | 0 0 0 0 0 0 0 0 2 1 0 2 2 1 2 2
%e 7 | 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 0
%e 8 | 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 0
%e 9 | 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 0
%e 10 | 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0
%e 11 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
%e 12 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
%e 13 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
%e 14 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
%e 15 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
%o (PARI) See Links section.
%Y Cf. A068601.
%K sign,tabl
%O 0,26
%A _Rémy Sigrist_, Apr 22 2023
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