%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