A348761 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the imaginary part of f(n) = Sum_{k >= 0} ((-1)^Sum_{j = 0..k-1} b_j) * (1+i)^k (where i denotes the imaginary unit); sequence A348760 gives the real part.

0, 0, 1, -1, 2, -2, -1, 1, 2, -2, -1, 1, 0, 0, 1, -1, 0, 0, 1, -1, 2, -2, -1, 1, 2, -2, -1, 1, 0, 0, 1, -1, -4, 4, 5, -5, 6, -6, -5, 5, 6, -6, -5, 5, -4, 4, 5, -5, 4, -4, -3, 3, -2, 2, 3, -3, -2, 2, 3, -3, 4, -4, -3, 3, -8, 8, 9, -9, 10, -10, -9, 9, 10, -10

Offset

0,5

Comments

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

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8191

Rémy Sigrist, Colored representation of f(n) for n < 2^20 in the complex plane (the hue is function of n)

Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of A000120(n) mod 2)

Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of the binary length of n, A070939(n))

Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of A000120(n), darker shades correspond to higher values)

Formula

a(2^k) = A009545(k) for any k >= 0.

Prog

(PARI) a(n) = { my (v=0, k, o=-1); while (n, n-=2^k=valuation(n, 2); v+=(1+I)^k * (-1)^o++); imag(v) }

Crossrefs

See A348691 for a similar sequence.

Cf. A000120, A009545, A070939, A348760.

Sequence in context: A214039 A089679 A290320 * A318703 A332384 A275885

Adjacent sequences: A348758 A348759 A348760 * A348762 A348763 A348764

Keyword

sign,look,base

Author

Rémy Sigrist, Oct 31 2021

Status

approved