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A348652 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k*13^k, a(n) is the real 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 A348653 for the imaginary part.

%I A348652 #14 Oct 29 2021 13:31:55
%S A348652 0,1,1,1,0,-1,-2,-1,-1,-1,0,1,2,3,4,4,4,3,2,1,2,2,2,3,4,5,1,2,2,2,1,0,
%T A348652 -1,0,0,0,1,2,3,-1,0,0,0,-1,-2,-3,-2,-2,-2,-1,0,1,-2,-1,-1,-1,-2,-3,
%U A348652 -4,-3,-3,-3,-2,-1,0,-5,-4,-4,-4,-5,-6,-7,-6,-6,-6
%N A348652 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k*13^k, a(n) is the real 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 A348653 for the imaginary part.
%C A348652 The function f defines a bijection from the nonnegative integers to the Gaussian integers.
%C A348652 The following diagram depicts g(d) for d = 0..12:
%C A348652                       |
%C A348652                       |    +
%C A348652                       |     3
%C A348652                       |
%C A348652             +    +    +    +
%C A348652              6    5   |4    2
%C A348652                       |
%C A348652          --------+----+----+-------
%C A348652                   7   |0    1
%C A348652                       |
%C A348652                  +    +    +    +
%C A348652                   8   |10   11   12
%C A348652                       |
%C A348652                  +    |
%C A348652                   9   |
%H A348652 Rémy Sigrist, <a href="/A348652/b348652.txt">Table of n, a(n) for n = 0..2197</a>
%H A348652 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 A348652 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 A348652 a(13^k) = A121622(k).
%o A348652 (PARI) g(d) = { if (d==0, 0, (1+I*((d-1)%3))*I^((d-1)\3)) }
%o A348652 a(n) = real(subst(Pol([g(d)|d<-digits(n, 13)]), 'x, 3+2*I))
%Y A348652 See A316657 for a similar sequence.
%Y A348652 Cf. A121622, A348653.
%K A348652 sign,base
%O A348652 0,7
%A A348652 _Rémy Sigrist_, Oct 27 2021