A322574 - OEIS

%I #16 Aug 27 2024 18:30:18

%S 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,

%T -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,

%U -1,-5,-1,7,-4,-15,-4,4,-1,-6,-1,8,3,-2,3,8,0,-8,0

%N 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).

%C Will z run through every Gaussian integer?

%H Rémy Sigrist, <a href="/A322574/b322574.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A322574/a322574.png">Colored representation of z(n) for n = 1..400000 in the complex plane</a> (where the hue is function of n)

%H Rémy Sigrist, <a href="/A322574/a322574_1.png">Colored representation of z(n) such that max(|Re(z(n))|, |Im(z(n))|) < 1000 for n = 1..10000000 in the complex plane</a> (where the hue is function of n)

%H Rémy Sigrist, <a href="/A322574/a322574.gp.txt">PARI program for A322574</a>

%e The first terms, alongside z(n), k(n) and associate children, are:

%e   n   a(n)  z(n)     k  z(4*n-2)  z(4*n-1)  z(4*n)  z(4*n+1)

%e   --  ----  -------  -  --------  --------  ------  --------

%e    1     0        0  1         1         i      -1        -i

%e    2     1        1  3         4   1 + 3*i      -2   1 - 3*i

%e    3     0        i  3     3 + i       4*i  -3 + i      -2*i

%e    4    -1       -1  3         2  -1 + 3*i      -4  -1 - 3*i

%e    5     0       -i  3     3 - i       2*i  -3 - i      -4*i

%e    6     4        4  1         5     4 + i       3     4 - i

%e    7     1  1 + 3*i  1   2 + 3*i   1 + 4*i     3*i   1 + 2*i

%e    8    -2       -2  8         6  -2 + 8*i     -10  -2 - 8*i

%e    9     1  1 - 3*i  1   2 - 3*i   1 - 2*i    -3*i   1 - 4*i

%e   10     3    3 + i  4     7 + i   3 + 5*i  -1 + i   3 - 3*i

%o (PARI) \\ See Links section.

%Y See A322575 for the imaginary part of z.

%Y This sequence is a complex variant of A322510.

%K sign

%O 1,6

%A _Rémy Sigrist_, Dec 17 2018