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A348916 a(n) is the "real" part of f(n) = Sum_{k >= 0} g(d_k) * (4 + w)^k where g(0) = 0 and g(1 + u + 2*v) = (2 + w)^u * (1 + w)^v for any u = 0..1 and v = 0..5, Sum_{k >= 0} d_k * 13^k is the base-13 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A348917 gives "w" parts.

%I A348916 #18 Nov 09 2021 15:01:53
%S A348916 0,1,2,1,1,0,-1,-1,-2,-1,-1,0,1,4,5,6,5,5,4,3,3,2,3,3,4,5,7,8,9,8,8,7,
%T A348916 6,6,5,6,6,7,8,3,4,5,4,4,3,2,2,1,2,2,3,4,2,3,4,3,3,2,1,1,0,1,1,2,3,-1,
%U A348916 0,1,0,0,-1,-2,-2,-3,-2,-2,-1,0,-5,-4,-3
%N A348916 a(n) is the "real" part of f(n) = Sum_{k >= 0} g(d_k) * (4 + w)^k where g(0) = 0 and g(1 + u + 2*v) = (2 + w)^u * (1 + w)^v for any u = 0..1 and v = 0..5, Sum_{k >= 0} d_k * 13^k is the base-13 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A348917 gives "w" parts.
%C A348916 For any Eisenstein integer z = u + v*w (where u and v are integers), we call u the "real" part of z and v the "w" part of z.
%C A348916 This sequence combines features of A334492 and of A348652.
%C A348916 It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.
%C A348916 The following diagram depicts g(d) for d = 0..12:
%C A348916            "w" axis
%C A348916                \
%C A348916                 .     .
%C A348916                  \   4
%C A348916                   \
%C A348916              .     .     .     .
%C A348916             6     5 \   3     2
%C A348916                      \
%C A348916           ._____._____._____._____._ "real" axis
%C A348916                7     0 \   1
%C A348916                         \
%C A348916              .     .     .     .
%C A348916             8     9    11 \  12
%C A348916                            \
%C A348916                       .     .
%C A348916                     10       \
%H A348916 Rémy Sigrist, <a href="/A348916/b348916.txt">Table of n, a(n) for n = 0..2196</a>
%H A348916 Joerg Arndt, <a href="/A348916/a348916_1.png">Representation of a similar construction</a>
%H A348916 Rémy Sigrist, <a href="/A348916/a348916.png">Colored representation of f for n = 0..13^5-1 in the complex plane</a> (the hue is function of n)
%H A348916 Rémy Sigrist, <a href="/A348916/a348916.gp.txt">PARI program for A348916</a>
%H A348916 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>
%o A348916 (PARI) See Links section.
%Y A348916 Cf. A334492, A348652, A348917.
%K A348916 sign,base
%O A348916 0,3
%A A348916 _Rémy Sigrist_, Nov 03 2021