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

Author

Rémy Sigrist, Jun 22 2019

Keywords

• sign
• look

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.

Examples

			See representation of the first layers of the binary tree in links section.
		

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

Crossrefs

See A326281 for the imaginary part of f.

Cf. A000120, A023416, A322574.

Programs

PARI

See Links section.