

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) * (1i)^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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 2^201 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, Table of n, a(n) for n = 1..8191
Rémy Sigrist, Representation of the first layers of the binary tree
Rémy Sigrist, Colored representation of f(n) for n = 1..2^201 (where the hue is function of n)
Rémy Sigrist, Colored representation of f(n) for n = 1..2^201 (where black pixels correspond to even n)
Rémy Sigrist, Density plot of the first 2^221 terms
Rémy Sigrist, PARI program for A326280


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.
Cf. A000120, A023416, A322574.
Sequence in context: A285864 A092869 A029337 * A280817 A060086 A308680
Adjacent sequences: A326277 A326278 A326279 * A326281 A326282 A326283


KEYWORD

sign,look


AUTHOR

Rémy Sigrist, Jun 22 2019


STATUS

approved



