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A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k*13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)*(3+2*i)^k where g(0) = 0, and g(1+u+3*v) = (1+u*i)*i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.

%I A348653 #12 Oct 29 2021 13:32:03
%S A348653 0,0,1,2,1,1,1,0,-1,-2,-1,-1,-1,2,2,3,4,3,3,3,2,1,0,1,1,1,5,5,6,7,6,6,
%T A348653 6,5,4,3,4,4,4,8,8,9,10,9,9,9,8,7,6,7,7,7,3,3,4,5,4,4,4,3,2,1,2,2,2,1,
%U A348653 1,2,3,2,2,2,1,0,-1,0,0,0,-1,-1,0,1,0,0
%N A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k*13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)*(3+2*i)^k where g(0) = 0, and g(1+u+3*v) = (1+u*i)*i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.
%C A348653 The function f defines a bijection from the nonnegative integers to the Gaussian integers.
%C A348653 The following diagram depicts g(d) for d = 0..12:
%C A348653                       |
%C A348653                       |    +
%C A348653                       |     3
%C A348653                       |
%C A348653             +    +    +    +
%C A348653              6    5   |4    2
%C A348653                       |
%C A348653          --------+----+----+-------
%C A348653                   7   |0    1
%C A348653                       |
%C A348653                  +    +    +    +
%C A348653                   8   |10   11   12
%C A348653                       |
%C A348653                  +    |
%C A348653                   9   |
%H A348653 Rémy Sigrist, <a href="/A348653/b348653.txt">Table of n, a(n) for n = 0..2197</a>
%H A348653 Stephen K. Lucas, <a href="https://www.researchgate.net/publication/355680849_Base_2_i_with_digit_set_0_1_i">Base 2 + i with digit set {0, +/-1, +/-i}</a>, ResearchGate (October 2021).
%H A348653 Rémy Sigrist, <a href="/A348652/a348652.png">Colored representation of f for n = 0..13^5-1 in the complex plane</a> (the hue is function of n)
%F A348653 abs(a(13^k)) = A188982(k).
%o A348653 (PARI) g(d) = { if (d==0, 0, (1+I*((d-1)%3))*I^((d-1)\3)) }
%o A348653 a(n) = imag(subst(Pol([g(d)|d<-digits(n, 13)]), 'x, 3+2*I))
%Y A348653 See A316658 for a similar sequence.
%Y A348653 Cf. A188982, A348652.
%K A348653 sign,base
%O A348653 0,4
%A A348653 _Rémy Sigrist_, Oct 27 2021