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A290884 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 real part of the n-th term of S.

%I A290884 #28 Apr 08 2020 08:39:52
%S A290884 0,0,1,1,2,2,3,3,2,2,3,3,4,4,5,5,0,0,1,1,2,2,3,3,2,2,3,3,4,4,5,5,-4,
%T A290884 -4,-3,-3,-2,-2,-1,-1,-2,-2,-1,-1,0,0,1,1,-4,-4,-3,-3,-2,-2,-1,-1,-2,
%U A290884 -2,-1,-1,0,0,1,1,-8,-8,-7,-7,-6,-6,-5,-5,-6,-6,-5
%N A290884 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 real part of the n-th term of S.
%C A290884 See A290885 for the imaginary part of the n-th term of S.
%C A290884 See A290886 for the square of the norm of the n-th term of S.
%C A290884 This sequence is a variant of A290536.
%C A290884 The representation of the first terms of S in the complex plane has nice fractal features, and looks like a Dragon curve (see also Links section).
%C A290884 The building of this sequence is close to that of the Twindragon (see Wikipedia link).
%C A290884 The sequence S' built with the same rules but with the initial term S'(1) = 1 seems to be the complement of S; the set of elements of S is the image of the set of elements of S' with respect to the symmetry z -> 1 - z.
%C A290884 From _Rémy Sigrist_, Jul 10 2018: (Start)
%C A290884 For any n >= 0 with binary expansion Sum_{k=0..h} b_k * 2^k, let g(n) = Sum_{k=0..h} b_k * (1+i)^k (where i denotes the imaginary unit).
%C A290884 Apparently, g(n) = i * a(n+1) - A290885(n+1) for any n >= 0.
%C A290884 The function g has similarities with the function f defined in A316657.
%C A290884 (End)
%H A290884 Rémy Sigrist, <a href="/A290884/b290884.txt">Table of n, a(n) for n = 1..10000</a>
%H A290884 Rémy Sigrist, <a href="/A290884/a290884.png">Representation of the first 100000 terms of S in the complex plane</a>
%H A290884 Rémy Sigrist, <a href="/A290884/a290884_1.png">Colorized representation of the first 100000 terms of S in the complex plane</a>
%H A290884 Rémy Sigrist, <a href="/A290884/a290884_2.png">Colorized representation of the first 1000000 terms of S in the complex plane</a>
%H A290884 Rémy Sigrist, <a href="/A290884/a290884_3.png">Colorized representation of the first 100000 terms of S' in the complex plane</a>
%H A290884 Rémy Sigrist, <a href="/A290884/a290884.gp.txt">PARI program for A290884</a>
%H A290884 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dragon_curve">Dragon curve</a>
%H A290884 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dragon_curve#Twindragon">Twindragon</a>
%e A290884 Let f be the function z -> z * (1+i), and g the function z -> (z-1) * (1+i) + 1.
%e A290884 S(1) = 0 by definition; so a(1) = 0.
%e A290884 f(S(1)) = 0 has already occurred.
%e A290884 g(S(1)) = -i has not yet occurred; so S(2) = -i and a(2) = 0.
%e A290884 f(S(2)) = 1 - i has not yet occurred; so S(3) = 1 - i and a(3) = 1.
%e A290884 g(S(2)) = 1 - 2*i has not yet occurred; so S(4) = 1 - 2*i and a(4) = 1.
%e A290884 f(S(3)) = 2 has not yet occurred; so S(5) = 2 and a(5) = 2.
%e A290884 g(S(3)) = 2 - i has not yet occurred; so S(6) = 2 - i and a(6) = 2.
%e A290884 f(S(4)) = 3 - i has not yet occurred; so S(7) = 3 - i and a(7) = 3.
%e A290884 g(S(4)) = 3 - 2*i has not yet occurred; so S(8) = 3 - 2*i and a(8) = 3.
%o A290884 (PARI) See Links section.
%o A290884 (PARI) a(n) = imag(subst(Pol(binary(n-1)),'x,I+1)); \\ _Kevin Ryde_, Apr 04 2020
%Y A290884 Cf. A290536, A290885, A290886, A316657.
%K A290884 sign,base,look
%O A290884 1,5
%A A290884 _Rémy Sigrist_, Aug 13 2017