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%I #22 Dec 10 2019 03:29:10
%S 0,1,1,1,2,2,1,0,2,3,3,4,3,2,0,-1,0,2,3,3,4,4,3,4,5,4,3,4,2,0,-1,-2,
%T -2,-1,1,0,3,4,6,2,4,5,7,6,5,5,2,1,5,7,8,7,6,4,1,5,5,3,0,2,-1,-2,-2,
%U -2,-3,-3,-2,-3,-1,1,5,-2,0,3,6,4,6,7,6,0,2,4
%N Let f(n) be a sequence of distinct Gaussian integers such that f(1) = 0 and for any n > 1, f(n) = f(floor(n/2)) + k(n)*g((1+i)^(A000120(n)-1) * (1-i)^A023416(n)) where k(n) > 0 is as small as possible and g(z) = z/gcd(Re(z), Im(z)); a(n) is the real part of f(n).
%C The idea underlying this sequence is to build an infinite binary tree of Gaussian integers:
%C - for any n > 0, f(n) has children f(2*n) and f(2*n+1),
%C - f(n), f(2*n) and f(2*n+1) form a right triangle,
%C - when u has child v and v has child w, then the angle between the vectors (u,v) and (v,w) is 45 degrees.
%C Among the first 2^20-1 terms, some values around the origin are missing: -2 - 3*i, -2, i, 2 - 2*i, 2, 4 + i, 5 - 2*i; will they ever appear?
%C Graphically, f has interesting features (see representations of f in Links section).
%C This sequence has similarities with A322574.
%H Rémy Sigrist, <a href="/A326280/b326280.txt">Table of n, a(n) for n = 1..8191</a>
%H Rémy Sigrist, <a href="/A326280/a326280.png">Representation of the first layers of the binary tree</a>
%H Rémy Sigrist, <a href="/A326280/a326280_1.png">Colored representation of f(n) for n = 1..2^20-1</a> (where the hue is function of n)
%H Rémy Sigrist, <a href="/A326280/a326280_2.png">Colored representation of f(n) for n = 1..2^20-1</a> (where black pixels correspond to even n)
%H Rémy Sigrist, <a href="/A326280/a326280_3.png">Density plot of the first 2^22-1 terms</a>
%H Rémy Sigrist, <a href="/A326280/a326280.gp.txt">PARI program for A326280</a>
%e See representation of the first layers of the binary tree in links section.
%o (PARI) See Links section.
%Y See A326281 for the imaginary part of f.
%Y Cf. A000120, A023416, A322574.
%K sign,look
%O 1,5
%A _Rémy Sigrist_, Jun 22 2019
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