A290884 Let S be the sequence generated by these rules: 0 is in S, and if z is in S, then z * (1+i) and (z-1) * (1+i) + 1 are in S (where i denotes the imaginary unit), and duplicates are deleted as they occur; a(n) = the real part of the n-th term of S.

0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 5, 5, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 5, 5, -4, -4, -3, -3, -2, -2, -1, -1, -2, -2, -1, -1, 0, 0, 1, 1, -4, -4, -3, -3, -2, -2, -1, -1, -2, -2, -1, -1, 0, 0, 1, 1, -8, -8, -7, -7, -6, -6, -5, -5, -6, -6, -5

Offset: 1

Author

Rémy Sigrist, Aug 13 2017

Keywords

• sign
• base
• look

Comments

See A290885 for the imaginary part of the n-th term of S.

See A290886 for the square of the norm of the n-th term of S.

This sequence is a variant of A290536.

The representation of the first terms of S in the complex plane has nice fractal features, and looks like a Dragon curve (see also Links section).

The building of this sequence is close to that of the Twindragon (see Wikipedia link).

The sequence S' built with the same rules but with the initial term S'(1) = 1 seems to be the complement of S; the set of elements of S is the image of the set of elements of S' with respect to the symmetry z -> 1 - z.

From Rémy Sigrist, Jul 10 2018: (Start)

For any n >= 0 with binary expansion Sum_{k=0..h} b_k * 2^k, let g(n) = Sum_{k=0..h} b_k * (1+i)^k (where i denotes the imaginary unit).

Apparently, g(n) = i * a(n+1) - A290885(n+1) for any n >= 0.

The function g has similarities with the function f defined in A316657.

(End)

Examples

			Let f be the function z -> z * (1+i), and g the function z -> (z-1) * (1+i) + 1.
S(1) = 0 by definition; so a(1) = 0.
f(S(1)) = 0 has already occurred.
g(S(1)) = -i has not yet occurred; so S(2) = -i and a(2) = 0.
f(S(2)) = 1 - i has not yet occurred; so S(3) = 1 - i and a(3) = 1.
g(S(2)) = 1 - 2*i has not yet occurred; so S(4) = 1 - 2*i and a(4) = 1.
f(S(3)) = 2 has not yet occurred; so S(5) = 2 and a(5) = 2.
g(S(3)) = 2 - i has not yet occurred; so S(6) = 2 - i and a(6) = 2.
f(S(4)) = 3 - i has not yet occurred; so S(7) = 3 - i and a(7) = 3.
g(S(4)) = 3 - 2*i has not yet occurred; so S(8) = 3 - 2*i and a(8) = 3.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Representation of the first 100000 terms of S in the complex plane
• Rémy Sigrist, Colorized representation of the first 100000 terms of S in the complex plane
• Rémy Sigrist, Colorized representation of the first 1000000 terms of S in the complex plane
• Rémy Sigrist, Colorized representation of the first 100000 terms of S' in the complex plane
• Rémy Sigrist, PARI program for A290884
• Wikipedia, Dragon curve
• Wikipedia, Twindragon

Crossrefs

Cf. A290536, A290885, A290886, A316657.

Programs

PARI

See Links section.

PARI

a(n) = imag(subst(Pol(binary(n-1)),'x,I+1)); \\ Kevin Ryde, Apr 04 2020