Eisenstein-Fibonacci sequences

[Joerg Arndt]

* Jonathan Post <jvospost3 at gmail.com> [Nov 01. 2006 10:14]:
> 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.
> 
> [...]
> 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.

Ralf Stephan's ggf() says that the OGF may be:

? g=ggf([1, 1, 3, 1, 7, 4,14, 17, 28, 54, 66, 143, 182, 350])
(x^5 - 3*x^3 + x + 1)/(-x^6 + 3*x^4 - x^3 - 3*x^2 + 1)
? factor(g)

[x - 1 1]

[x^4 + x^3 - 2*x^2 - 2*x - 1 1]

[x^2 + x - 1 -1]

[x^4 - x^3 - x^2 + x + 1 -1]


> [...]