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

0,26

Comments

This sequence is related to the Newton fractal, and exhibits similar rich patterns (see illustration in Links section).

Links

Table of n, a(n) for n=0..90.

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

Adjacent sequences: A362493 A362494 A362495 * A362497 A362498 A362499

Keyword

sign,tabl

Author

Rémy Sigrist, Apr 22 2023

Status

approved