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A334492 a(n) is the "real" part of f(n) = Sum_{k>=0, d_k>0} (1+w)^(d_k-1) * (3+w)^k where Sum_{k>=0} d_k * 7^k is the base 7 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A334493 gives "w" parts.

%I A334492 #17 Jul 25 2021 03:36:43
%S A334492 0,1,1,0,-1,-1,0,3,4,4,3,2,2,3,2,3,3,2,1,1,2,-1,0,0,-1,-2,-2,-1,-3,-2,
%T A334492 -2,-3,-4,-4,-3,-2,-1,-1,-2,-3,-3,-2,1,2,2,1,0,0,1,8,9,9,8,7,7,8,11,
%U A334492 12,12,11,10,10,11,10,11,11,10,9,9,10,7,8,8,7,6,6
%N A334492 a(n) is the "real" part of f(n) = Sum_{k>=0, d_k>0} (1+w)^(d_k-1) * (3+w)^k where Sum_{k>=0} d_k * 7^k is the base 7 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A334493 gives "w" parts.
%C A334492 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 A334492 This sequence has connections with A316657; here we work with Eisenstein integers, there with Gaussian integers.
%C A334492 It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.
%H A334492 Rémy Sigrist, <a href="/A334492/b334492.txt">Table of n, a(n) for n = 0..16806</a>
%H A334492 Rémy Sigrist, <a href="/A334492/a334492.png">Colored representation of f(n) for n = 0..7^7-1 in a hexagonal lattice</a> (where the hue is function of n)
%H A334492 Rémy Sigrist, <a href="/A334492/a334492.gp.txt">PARI program for A334492</a>
%H A334492 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>
%e A334492 The following diagram depicts f(n) for n = 0..13:
%e A334492             "w" axis
%e A334492                 \
%e A334492            .     .     .     .     .     .     .     .
%e A334492                   \              10     9
%e A334492                    \
%e A334492         .     .     .     .     .     .     .     .
%e A334492                    3 \   2    11     7     8
%e A334492                       \
%e A334492            ._____._____._____._____._____._____._____. "real" axis
%e A334492                 4     0 \   1    12    13
%e A334492                          \
%e A334492         .     .     .     .     .     .     .     .
%e A334492                    5     6 \
%e A334492 - f(9) = 4 + 2*w, hence a(9) = 4.
%o A334492 (PARI) See Links section.
%Y A334492 Cf. A307013 (equivalent coordinate for a counterclockwise spiral), A316657, A334493.
%K A334492 sign,base,look
%O A334492 0,8
%A A334492 _Rémy Sigrist_, May 03 2020