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A326280
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).
2
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, -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, -2, -3, -3, -2, -3, -1, 1, 5, -2, 0, 3, 6, 4, 6, 7, 6, 0, 2, 4
OFFSET
1,5
COMMENTS
The idea underlying this sequence is to build an infinite binary tree of Gaussian integers:
- for any n > 0, f(n) has children f(2*n) and f(2*n+1),
- f(n), f(2*n) and f(2*n+1) form a right triangle,
- when u has child v and v has child w, then the angle between the vectors (u,v) and (v,w) is 45 degrees.
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?
Graphically, f has interesting features (see representations of f in Links section).
This sequence has similarities with A322574.
LINKS
Rémy Sigrist, Colored representation of f(n) for n = 1..2^20-1 (where the hue is function of n)
Rémy Sigrist, Colored representation of f(n) for n = 1..2^20-1 (where black pixels correspond to even n)
EXAMPLE
See representation of the first layers of the binary tree in links section.
PROG
(PARI) See Links section.
CROSSREFS
See A326281 for the imaginary part of f.
Sequence in context: A285864 A092869 A029337 * A350310 A280817 A060086
KEYWORD
sign,look
AUTHOR
Rémy Sigrist, Jun 22 2019
STATUS
approved