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A348917 a(n) is the "w" 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 A348916 gives "real" parts.

%I A348917 #15 Nov 09 2021 15:02:00
%S A348917 0,0,1,1,2,1,1,0,-1,-1,-2,-1,-1,1,1,2,2,3,2,2,1,0,0,-1,0,0,5,5,6,6,7,
%T A348917 6,6,5,4,4,3,4,4,4,4,5,5,6,5,5,4,3,3,2,3,3,7,7,8,8,9,8,8,7,6,6,5,6,6,
%U A348917 3,3,4,4,5,4,4,3,2,2,1,2,2,2,2,3,3,4,3,3
%N A348917 a(n) is the "w" 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 A348916 gives "real" parts.
%C A348917 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 A348917 This sequence combines features of A334493 and of A348653.
%C A348917 It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.
%C A348917 The following diagram depicts g(d) for d = 0..12:
%C A348917            "w" axis
%C A348917                \
%C A348917                 .     .
%C A348917                  \   4
%C A348917                   \
%C A348917              .     .     .     .
%C A348917             6     5 \   3     2
%C A348917                      \
%C A348917           ._____._____._____._____._ "real" axis
%C A348917                7     0 \   1
%C A348917                         \
%C A348917              .     .     .     .
%C A348917             8     9    11 \  12
%C A348917                            \
%C A348917                       .     .
%C A348917                     10       \
%H A348917 Rémy Sigrist, <a href="/A348917/b348917.txt">Table of n, a(n) for n = 0..2196</a>
%H A348917 Joerg Arndt, <a href="/A348916/a348916_1.png">Representation of a similar construction</a>
%H A348917 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 A348917 Rémy Sigrist, <a href="/A348917/a348917.gp.txt">PARI program for A348917</a>
%H A348917 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>
%o A348917 (PARI) See Links section.
%Y A348917 Cf. A334493, A348653, A348916.
%K A348917 sign,base
%O A348917 0,5
%A A348917 _Rémy Sigrist_, Nov 03 2021