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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.

%I A332412 #18 Feb 15 2020 10:43:55
%S A332412 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,
%T A332412 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,
%U A332412 0,1,0,-1,0,-3,-2,-3,-4,-3,0,1,0,-1,0,-9,-8,-9
%N 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.
%C A332412 The representation of {f(n)} corresponds to the cross form of the Vicsek fractal.
%C A332412 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.
%H A332412 Rémy Sigrist, <a href="/A332412/b332412.txt">Table of n, a(n) for n = 0..15624</a>
%H A332412 Rémy Sigrist, <a href="/A332412/a332412.png">Colored representation of f(n) for n = 0..5^6-1 in the complex plan</a> (where the hue is function of n)
%H A332412 Wikipedia, <a href="https://en.wikipedia.org/wiki/Vicsek_fractal">Vicsek fractal</a>
%F A332412 a(n) = 0 iff the n-th row of A031219 has only even terms.
%F A332412 a(5*n)   = 3*a(n).
%F A332412 a(5*n+1) = 3*a(n) + 1.
%F A332412 a(5*n+2) = 3*a(n).
%F A332412 a(5*n+3) = 3*a(n) - 1.
%F A332412 a(5*n+4) = 3*a(n).
%e A332412 For n = 103:
%e A332412 - 103 = 4*5^2 + 3*5^0,
%e A332412 - so f(123) = 3^2 * i^(4-1) + 3^0 * i^(3-1) = -1 - 9*i,
%e A332412 - and a(n) = -1.
%o A332412 (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))) }
%Y A332412 See A332497 for a similar sequence.
%Y A332412 Cf. A031219, A289813, A332413 (imaginary parts).
%K A332412 sign,base
%O A332412 0,6
%A A332412 _Rémy Sigrist_, Feb 12 2020