A318479 -id:A318479 - cp's OEIS frontend

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

Rémy Sigrist, Aug 26 2018

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.

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

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

Cf. A316657 (base 2+i).

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

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

A330714 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^k (where i denotes the imaginary unit); a(n) is the square of the modulus of h(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 2, 1, 1, 2, 0, 1, 2, 1, 1, 0, 1, 4, 2, 5, 0, 1, 1, 2, 2, 5, 1, 4, 1, 2, 0, 1, 1, 2, 4, 5, 2, 1, 5, 4, 0, 1, 1, 2, 1, 0, 2, 1, 2, 5, 5, 8, 1, 2, 4, 5, 1, 4, 2, 5, 0, 1, 1, 2, 1, 0, 2, 1, 4, 1, 5, 2, 2, 1, 1, 0, 5, 2, 4, 1, 0, 1, 1, 2, 1

Offset: 0

Seiichi Manyama, Dec 27 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A007088, A131851, A131852, A131853, A131856, A131858, A131860, A290886, A318479.

a[0] = 0; a[n_] := a[n] = a[Floor[n/2]]*I + Mod[n, 2]; Table[Abs[a[n]]^2, {n, 0, 100}] (* Amiram Eldar, May 06 2021, after Jean-François Alcover at A131851 *)

{a(n) = my(d=Vecrev(digits(n, 2))); norm(sum(k=1, #d, d[k]*I^k))}

a(n) = A131851(n)^2 + A131852(n)^2.

cp's OEIS Frontend

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).

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