

A332412


a(n) is the real part of f(n) = Sum_{d_k > 0} 3^k * i^(d_k1) 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.


3



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
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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..5^61 in the complex plan (where the hue is function of n)
Wikipedia, Vicsek fractal


FORMULA

a(n) = 0 iff the nth 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^(41) + 3^0 * i^(31) = 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^(k1)*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



