A318438 For any n >= 0 with binary expansion Sum_{k=0..w} b_k * 2^k, let h(n) = Sum_{k=0..w} b_k * (i-1)^k (where i denotes the imaginary unit); a(n) is the real part of h(n).

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

Offset: 0

Author

Rémy Sigrist, Aug 26 2018

Keywords

• sign
• look
• base

Comments

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.

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, Colored scatterplot of (a(n), A318439(n)) for n = 0..2^20-1 (where the hue is function of n)
• Wikipedia, Base -1+/-i
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A009116, A318439 (imaginary part), A318479 (norm), A340669 (negation).

Cf. A316657 (base 2+i).

Programs

PARI

a(n) = my (d=Vecrev(digits(n,2))); real(sum(i=1, #d, d[i]*(I-1)^(i-1)))

Formula

a(2^k) = A009116(k) for any k >= 0.