A322574 z(1) = 0, and for any n > 0, z(4*n-2) = z(n) + k(n), z(4*n-1) = z(n) + i*k(n), z(4*n) = z(n) - k(n) and z(4*n+1) = z(n) - i*k(n) where k(n) is the least positive integer not leading to a duplicate term in sequence z (and i denotes the imaginary unit); a(n) is the real part of z(n).

0, 1, 0, -1, 0, 4, 1, -2, 1, 3, 0, -3, 0, 2, -1, -4, -1, 3, 0, -3, 0, 5, 4, 3, 4, 2, 1, 0, 1, 6, -2, -10, -2, 2, 1, 0, 1, 7, 3, -1, 3, 9, 0, -9, 0, -2, -3, -4, -3, 7, 0, -7, 0, 9, 2, -5, 2, 3, -1, -5, -1, 7, -4, -15, -4, 4, -1, -6, -1, 8, 3, -2, 3, 8, 0, -8, 0

Offset: 1

Author

Rémy Sigrist, Dec 17 2018

Keywords

• sign

Comments

Will z run through every Gaussian integer?

Examples

			The first terms, alongside z(n), k(n) and associate children, are:
  n   a(n)  z(n)     k  z(4*n-2)  z(4*n-1)  z(4*n)  z(4*n+1)
  --  ----  -------  -  --------  --------  ------  --------
   1     0        0  1         1         i      -1        -i
   2     1        1  3         4   1 + 3*i      -2   1 - 3*i
   3     0        i  3     3 + i       4*i  -3 + i      -2*i
   4    -1       -1  3         2  -1 + 3*i      -4  -1 - 3*i
   5     0       -i  3     3 - i       2*i  -3 - i      -4*i
   6     4        4  1         5     4 + i       3     4 - i
   7     1  1 + 3*i  1   2 + 3*i   1 + 4*i     3*i   1 + 2*i
   8    -2       -2  8         6  -2 + 8*i     -10  -2 - 8*i
   9     1  1 - 3*i  1   2 - 3*i   1 - 2*i    -3*i   1 - 4*i
  10     3    3 + i  4     7 + i   3 + 5*i  -1 + i   3 - 3*i
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored representation of z(n) for n = 1..400000 in the complex plane (where the hue is function of n)
• Rémy Sigrist, Colored representation of z(n) such that max(|Re(z(n))|, |Im(z(n))|) < 1000 for n = 1..10000000 in the complex plane (where the hue is function of n)
• Rémy Sigrist, PARI program for A322574

Crossrefs

See A322575 for the imaginary part of z.

This sequence is a complex variant of A322510.

Programs

PARI

\\ See Links section.