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, 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
Examples
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
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
Crossrefs
Cf. A068601.
Programs
-
PARI
See Links section.
Comments