A332412 a(n) is the real part of f(n) = Sum_{d_k > 0} 3^k * i^(d_k-1) where Sum_{k >= 0} 5^k * d_k is the base 5 representation of n and i denotes the imaginary unit. Sequence A332413 gives imaginary parts.

0, 1, 0, -1, 0, 3, 4, 3, 2, 3, 0, 1, 0, -1, 0, -3, -2, -3, -4, -3, 0, 1, 0, -1, 0, 9, 10, 9, 8, 9, 12, 13, 12, 11, 12, 9, 10, 9, 8, 9, 6, 7, 6, 5, 6, 9, 10, 9, 8, 9, 0, 1, 0, -1, 0, 3, 4, 3, 2, 3, 0, 1, 0, -1, 0, -3, -2, -3, -4, -3, 0, 1, 0, -1, 0, -9, -8, -9

Offset

0,6

Comments

The representation of {f(n)} corresponds to the cross form of the Vicsek fractal.

As a set, {f(n)} corresponds to the Gaussian integers whose real and imaginary parts have not simultaneously a nonzero digit at the same place in their balanced ternary representations.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..15624

Rémy Sigrist, Colored representation of f(n) for n = 0..5^6-1 in the complex plan (where the hue is function of n)

Wikipedia, Vicsek fractal

Formula

a(n) = 0 iff the n-th row of A031219 has only even terms.

a(5*n) = 3*a(n).

a(5*n+1) = 3*a(n) + 1.

a(5*n+2) = 3*a(n).

a(5*n+3) = 3*a(n) - 1.

a(5*n+4) = 3*a(n).

Example

For n = 103:
- 103 = 4*5^2 + 3*5^0,
- so f(123) = 3^2 * i^(4-1) + 3^0 * i^(3-1) = -1 - 9*i,
- and a(n) = -1.

Prog

(PARI) a(n) = { my (d=Vecrev(digits(n, 5))); real(sum (k=1, #d, if (d[k], 3^(k-1)*I^(d[k]-1), 0))) }

Crossrefs

See A332497 for a similar sequence.

Cf. A031219, A289813, A332413 (imaginary parts).

Sequence in context: A308430 A280136 A258451 * A333229 A164358 A275638

Adjacent sequences: A332409 A332410 A332411 * A332413 A332414 A332415

Keyword

sign,base

Author

Rémy Sigrist, Feb 12 2020

Status

approved