A316657 For any n >= 0 with base-5 expansion Sum_{k=0..w} d_k * 5^k, let f(n) = Sum_{k=0..w} [d_k > 0] * (2 + i)^k * i^(d_k - 1) (where [] is an Iverson bracket and i denotes the imaginary unit); a(n) equals the real part of f(n).

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

Offset: 0

Author

Rémy Sigrist, Jul 09 2018

Keywords

• sign
• base

Comments

See A316658 for the imaginary part of f.

See A316707 for the square of the modulus of f.

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

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

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..15624
• Rémy Sigrist, Colored scatterplot of (a(n), A316658(n)) for n=0..5^8-1 (where the hue is function of n)
• Wikipedia, Gaussian integer

Crossrefs

Cf. A139011, A316658, A316707.

Programs

Mathematica

a[n_] := Module[{d, z}, d = IntegerDigits[n, 5] // Reverse; z = Sum[ If[d[[i]]>0, (2+I)^(i-1)*I^(d[[i]]-1), 0], {i, 1, Length[d]}]; Re[z]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 06 2021, after PARI code *)

PARI

a(n) = my (d=Vecrev(digits(n, 5)), z=sum(i=1, #d, if (d[i], (2+I)^(i-1) * I^(d[i]-1), 0))); real(z)

Formula

a(5^n) = A139011(n) for any n >= 0.

a(3 * 5^n) = -A139011(n) for any n >= 0.