Eisenstein-Fibonacci sequences

[Jonathan Post]

Eisenstein-Fibonacci sequences.  Cube root of unity
analogues of square-root of unity A014291  Imaginary
Rabbits.

Let b(0) = w, b(1) = w^2, b(n) = w*b(n-1) + b(n-2),
where w = omega = (-1 + i*sqrt(3))/2 and w^2 = omega^2
= (-1 - i*sqrt(3))/2.

There are various integer sequences derived from this
complex sequence (such as the real part, the imaginary
part, the polynomial coefficient is of 1, w, and
w^2)Unless my arithmetic by hand is in error again,
is:

n   b(n)
0   w
1   w^2
2   w(w^2) + w = w^3 + w = 1 + w.
3   w(1 + w) + w^2 = w + w^2 + w^2 = 1 + 2w^2.
4   w(1 + 2w^2) + 1 + w = w + 2w^3 + 1 + w = 3 + 2w.
5   w(3 + 2w) + 1 + 2w^2 = 3w + 2w^2 + 1 + 2w^2.
   = 1 + 3w + 4w^2.
6  w(1 + 3w + 4w^2) + 3 + 2w = w + 3w^2 + 4w^3 + 3 +
2w
   = 7 + 3w + 3w^2.
7  w(7 + 3w + 3w^2) + 1 + 3w + 4w^2 = 7w + 3w^2 + 3w^3

  + 1 + 3w + 4w^2 = 4 + 10w + 7w^2.
8  w(4 + 10w + 7w^2) + 7 + 3w + 3w^2 = 4w + 10w^2 +
  7w^3 + 7 + 3w + 3w^2 = 14 + 7w + 13w^2.
9  w(14 + 7w + 13w^2) + 4 + 10w + 7w^2 = 14w + 7w^2 +
  13w^3 + 4 + 10w + 7w^2 = 17 + 24w + 14w^2.
10 w(17 + 24w + 14w^2) + 14 + 7w + 13w^2 = 17w + 24w^2
  + 14w^3 + 14 + 7w + 13w^2 = 28 + 24w + 37w^2.
11 w(28 + 24w + 37w^2) + 17 + 24w + 14w^2 = 28w +
  24w^2 + 37w^3 + 17 + 24w + 14w^2 = 54 + 52w +
38w^2.
12 w(54 + 52w + 38w^2) + 28 + 24w + 37w^2 = 54w +
  52w^2 + 38w^3 + 28 + 24w + 37w^2 = 66 + 78w +
89w^2.
13 w(66 + 78w + 89w^2) + 54 + 52w + 38w^2 =
  66w + 78w^2 + 89w^3 + 54 + 52w + 38w^2 =
  143 + 118w + 116w^2.
14 w(143 + 118w + 116w^2) + 66 + 78w + 89w^2 =
  143w + 118w^2 + 116w^3 + 66 + 78w + 89w^2 =
  182 + 221w + 207w^2.
15 w(182 + 221w + 207w^2) + 143 + 118w + 116w^2 =
  182w + 221w^2 + 207w^3 + 143 + 118w + 116w^2 =
   350 + 300w + 337w^2.

The coefficients of 1 = a(n) = 0, 0, 1, 1, 3, 1, 7, 4,
14, 17, 28, 54, 66, 143, 182, 350, ...

which does not seem to be in OEIS.

To excerpt my comment in A107890 (and some other
seqs):

"Eisenstein integers are of the form a + b*omega,
where a and b are ordinary integers, and omega = (-1 +
i*sqrt(3))/2 is a cube root of 1, the other cube roots
of 1 being 1 and omega^2 = (-1 - i*sqrt(3))/2.
Eisenstein integers are complex numbers that are also
members of the imaginary quadratic field Q(sqrt -3) =
Z[omega]. The sums, differences, and products of
Eisenstein integers are other Eisenstein integers."

Other sequences come from:
b(0) = w, b(1) = w^2, b(n) = b(n-1) + w*b(n-2),
and from the O, J notation of Dörrie (1965) in
Weisstein, Eric W. "Eisenstein Integer." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/EisensteinInteger.html
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