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A362496
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).
1
-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, 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, 0, 0, 0, 0, 0, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 1
OFFSET
0,26
COMMENTS
This sequence is related to the Newton fractal, and exhibits similar rich patterns (see illustration in Links section).
LINKS
Rémy Sigrist, Colored representation of the square array for n, k <= 1000 (black, white, blue and red pixels denote, respectively, -1, 0, 1 and 2)
Rémy Sigrist, PARI program
Wikipedia, Newton fractal
Wikipedia, Newton's method
EXAMPLE
Array A(n, k) begins:
n\k | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
----+------------------------------------------------------
0 | -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 | 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1
2 | 0 0 0 0 2 1 1 1 1 1 1 1 1 1 1 1
3 | 0 0 0 0 0 2 2 1 1 1 1 1 1 1 1 1
4 | 0 0 0 0 0 0 1 2 2 1 2 1 1 1 1 1
5 | 0 0 0 0 0 0 0 0 2 2 0 1 1 1 1 1
6 | 0 0 0 0 0 0 0 0 2 1 0 2 2 1 2 2
7 | 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 0
8 | 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 0
9 | 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 0
10 | 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0
11 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
12 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
13 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
14 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
15 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
PROG
(PARI) See Links section.
CROSSREFS
Cf. A068601.
Sequence in context: A309142 A064532 A321922 * A360003 A287146 A025926
KEYWORD
sign,tabl
AUTHOR
Rémy Sigrist, Apr 22 2023
STATUS
approved