A348916 - OEIS

%I #18 Nov 09 2021 15:01:53

%S 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 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 0,1,0,0,-1,-2,-2,-3,-2,-2,-1,0,-5,-4,-3

%N 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 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 This sequence combines features of A334492 and of A348652.

%C It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.

%C The following diagram depicts g(d) for d = 0..12:

%C            "w" axis

%C                \

%C                 .     .

%C                  \   4

%C                   \

%C              .     .     .     .

%C             6     5 \   3     2

%C                      \

%C           ._____._____._____._____._ "real" axis

%C                7     0 \   1

%C                         \

%C              .     .     .     .

%C             8     9    11 \  12

%C                            \

%C                       .     .

%C                     10       \

%H Rémy Sigrist, <a href="/A348916/b348916.txt">Table of n, a(n) for n = 0..2196</a>

%H Joerg Arndt, <a href="/A348916/a348916_1.png">Representation of a similar construction</a>

%H 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 Rémy Sigrist, <a href="/A348916/a348916.gp.txt">PARI program for A348916</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>

%o (PARI) See Links section.

%Y Cf. A334492, A348652, A348917.

%K sign,base

%O 0,3

%A _Rémy Sigrist_, Nov 03 2021