cp's OEIS Frontend

See A318439 for the imaginary part of h.

See A318479 for the square of the modulus of h.

The function h corresponds to the interpretation of the binary representation of a number in base -1+i and defines a bijection from the nonnegative integers to the Gaussian integers.

The function h has nice fractal features (see scatterplot in Links section).

This sequence has similarities with A316657.

For any Eisenstein integer z = u + v*w (where u and v are integers), we call u the "real" part of z and v the "w" part of z.

This sequence has connections with A316657; here we work with Eisenstein integers, there with Gaussian integers.

It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.

See A316657 for the real part of f and additional 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)

See A318703 for the imaginary part of f.

See A318704 for the square of the modulus of f.

The function f defines a bijection from the nonnegative integers to the Gaussian integers.

This sequence has similarities with A316657.

See A318706 for the imaginary part of g.

See A318707 for the square of the modulus of g.

The following diagrams shows s(k) for k = 0..8 in the complex plane:

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s(4) s(3) s(2)

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---s(5)--s(0)--s(1)---

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s(6) s(7) s(8)

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The function g defines a bijection from the nonnegative integers to the Gaussian integers.

This sequence has similarities with A316657.

See A318723 for the imaginary part of g.

See A318724 for the square of the modulus of g.

This sequence can be computed by considering the base 4 representation of A042968, hence the keyword base.

The function g runs uniquely through the set of Gaussian integers z such that Re(z) < 0 or Im(z) < 0.

The function g is related to the numbering of the cells in a Chair tiling (see representation of g(n) in Links section).

This sequence has similarities with A316657.

The function f defines a bijection from the nonnegative integers to the Gaussian integers.

The following diagram depicts g(d) for d = 0..12:

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| +

| 3

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+ + + +

6 5 |4 2

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--------+----+----+-------

7 |0 1

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+ + + +

8 |10 11 12

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+ |

9 |

See A316657 for the real part of f and additional comments.

This sequence is a self-inverse permutation of the nonnegative integers.

It is possible to build a similar sequence for any fixed base b > 1 and any permutation p of {1, ..., b-1}.

This sequence is interesting as it satisfies f(a(n)) = -f(n), where f(n) = (A316657(n), A316658(n)).