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%I #16 May 07 2020 19:38:18
%S 0,-1,-1,-2,0,-1,-1,-2,2,1,1,0,2,1,1,0,4,3,3,2,4,3,3,2,6,5,5,4,6,5,5,
%T 4,4,3,3,2,4,3,3,2,6,5,5,4,6,5,5,4,8,7,7,6,8,7,7,6,10,9,9,8,10,9,9,8,
%U 0,-1,-1,-2,0,-1,-1,-2,2,1,1,0,2,1,1,0,4,3
%N Let S be the sequence generated by these rules: 0 is in S, and if z is in S, then z * (1+i) and (z-1) * (1+i) + 1 are in S (where i denotes the imaginary unit), and duplicates are deleted as they occur; a(n) = the imaginary part of the n-th term of S.
%C See A290884 for the real part of the n-th term of S, and additional comments.
%C See A290886 for the square of the norm of the n-th term of S.
%H Rémy Sigrist, <a href="/A290885/b290885.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A290885/a290885.gp.txt">PARI program for A290885</a>
%e Let f be the function z -> z * (1+i), and g the function z -> (z-1) * (1+i) + 1.
%e S(1) = 0 by definition; so a(1) = 0.
%e f(S(1)) = 0 has already occurred.
%e g(S(1)) = -i has not yet occurred; so S(2) = -i and a(2) = -1.
%e f(S(2)) = 1 - i has not yet occurred; so S(3) = 1 - i and a(3) = -1.
%e g(S(2)) = 1 - 2*i has not yet occurred; so S(4) = 1 - 2*i and a(4) = -2.
%e f(S(3)) = 2 has not yet occurred; so S(5) = 2 and a(5) = 0.
%e g(S(3)) = 2 - i has not yet occurred; so S(6) = 2 - i and a(6) = -1.
%e f(S(4)) = 3 - i has not yet occurred; so S(7) = 3 - i and a(7) = -1.
%e g(S(4)) = 3 - 2*i has not yet occurred; so S(8) = 3 - 2*i and a(8) = -2.
%o (PARI) See Links section.
%o (PARI) a(n) = -real(subst(Pol(binary(n-1)),'x,I+1)); \\ _Kevin Ryde_, Apr 08 2020
%Y Cf. A290537, A290884, A290886.
%K sign,look
%O 1,4
%A _Rémy Sigrist_, Aug 13 2017
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