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A348920 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) = (1 + 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 A348921 gives "w" parts.

%I A348920 #18 Nov 09 2021 15:02:10
%S A348920 0,1,2,1,2,0,0,-1,-2,-1,-2,0,0,4,5,6,5,6,4,4,3,2,3,2,4,4,8,9,10,9,10,
%T A348920 8,8,7,6,7,6,8,8,3,4,5,4,5,3,3,2,1,2,1,3,3,6,7,8,7,8,6,6,5,4,5,4,6,6,
%U A348920 -1,0,1,0,1,-1,-1,-2,-3,-2,-3,-1,-1,-2,-1,0
%N A348920 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) = (1 + 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 A348921 gives "w" parts.
%C A348920 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 A348920 This sequence is a variant of A334492 and of A348916.
%C A348920 It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.
%C A348920 The following diagram depicts g(d) for d = 0..12:
%C A348920            "w" axis
%C A348920                \
%C A348920                 .     .     .
%C A348920                6 \         4
%C A348920                   \
%C A348920                    .     .
%C A348920                   5 \   3
%C A348920                      \
%C A348920           ._____._____._____._____._ "real" axis
%C A348920          8     7     0 \   1     2
%C A348920                         \
%C A348920                    .     .
%C A348920                   9    11 \
%C A348920                            \
%C A348920                 .     .     .
%C A348920               10          12 \
%H A348920 Rémy Sigrist, <a href="/A348920/b348920.txt">Table of n, a(n) for n = 0..2196</a>
%H A348920 Joerg Arndt, <a href="/A348920/a348920_1.png">Representation of a similar construction</a>
%H A348920 Rémy Sigrist, <a href="/A348920/a348920.png">Colored representation of f for n = 0..13^5-1 in the complex plane</a> (the hue is function of n)
%H A348920 Rémy Sigrist, <a href="/A348920/a348920.gp.txt">PARI program for A348920</a>
%H A348920 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>
%o A348920 (PARI) See Links section.
%Y A348920 Cf. A334492, A348916, A348921.
%K A348920 sign,base
%O A348920 0,3
%A A348920 _Rémy Sigrist_, Nov 04 2021