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).

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, 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..2^20-1 (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)

Rémy Sigrist, Density plot of the first 2^22-1 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 * A350310 A280817 A060086

Adjacent sequences: A326277 A326278 A326279 * A326281 A326282 A326283

Keyword

sign,look

Author

Rémy Sigrist, Jun 22 2019

Status

approved