cp's OEIS Frontend

A334493 a(n) is the "w" 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 A334492 gives "real" parts.

%I A334493 #13 Jul 31 2021 14:40:37
%S A334493 0,0,1,1,0,-1,-1,1,1,2,2,1,0,0,3,3,4,4,3,2,2,2,2,3,3,2,1,1,-1,-1,0,0,
%T A334493 -1,-2,-2,-3,-3,-2,-2,-3,-4,-4,-2,-2,-1,-1,-2,-3,-3,5,5,6,6,5,4,4,6,6,
%U A334493 7,7,6,5,5,8,8,9,9,8,7,7,7,7,8,8,7,6,6,4,4,5
%N A334493 a(n) is the "w" 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 A334492 gives "real" parts.
%C A334493 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 A334493 This sequence has connections with A316658; here we work with Eisenstein integers, there with Gaussian integers.
%C A334493 It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.
%H A334493 Rémy Sigrist, <a href="/A334493/b334493.txt">Table of n, a(n) for n = 0..16806</a>
%H A334493 Rémy Sigrist, <a href="/A334493/a334493.gp.txt">PARI program for A334493</a>
%H A334493 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>
%e A334493 The following diagram depicts f(n) for n = 0..13:
%e A334493             "w" axis
%e A334493                 \
%e A334493            .     .     .     .     .     .     .     .
%e A334493                   \              10     9
%e A334493                    \
%e A334493         .     .     .     .     .     .     .     .
%e A334493                    3 \   2    11     7     8
%e A334493                       \
%e A334493            ._____._____._____._____._____._____._____. "real" axis
%e A334493                 4     0 \   1    12    13
%e A334493                          \
%e A334493         .     .     .     .     .     .     .     .
%e A334493                    5     6 \
%e A334493 - f(9) = 4 + 2*w, hence a(9) = 2.
%o A334493 (PARI) See Links section.
%Y A334493 Cf. A307012 (equivalent coordinate for a counterclockwise spiral), A316658, A334492.
%K A334493 sign,base,look
%O A334493 0,10
%A A334493 _Rémy Sigrist_, May 03 2020