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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.
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%I #18 Nov 09 2021 15:02:10

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

%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) = (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 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 is a variant of A334492 and of A348916.

%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 6 \ 4

%C \

%C . .

%C 5 \ 3

%C \

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

%C 8 7 0 \ 1 2

%C \

%C . .

%C 9 11 \

%C \

%C . . .

%C 10 12 \

%H Rémy Sigrist, <a href="/A348920/b348920.txt">Table of n, a(n) for n = 0..2196</a>

%H Joerg Arndt, <a href="/A348920/a348920_1.png">Representation of a similar construction</a>

%H 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 Rémy Sigrist, <a href="/A348920/a348920.gp.txt">PARI program for A348920</a>

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

%o (PARI) See Links section.

%Y Cf. A334492, A348916, A348921.

%K sign,base

%O 0,3

%A _Rémy Sigrist_, Nov 04 2021