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A348916
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.
3
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, 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, 0, 1, 0, 0, -1, -2, -2, -3, -2, -2, -1, 0, -5, -4, -3
OFFSET
0,3
COMMENTS
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.
This sequence combines features of A334492 and of A348652.
It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.
The following diagram depicts g(d) for d = 0..12:
"w" axis
\
. .
\ 4
\
. . . .
6 5 \ 3 2
\
._____._____._____._____._ "real" axis
7 0 \ 1
\
. . . .
8 9 11 \ 12
\
. .
10 \
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
sign,base
AUTHOR
Rémy Sigrist, Nov 03 2021
STATUS
approved