Eisenstein-Fibonacci sequences

[Max A.]

On 10/22/06, Jonathan Post <jvospost3 at gmail.com> wrote:
> 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.

First off, using the identity w^2 = -w-1, we can uniquely represent
b(n) as u(n)+v(n)*w where u(n) and v(n) are integers.
In particular, u(0)=0, v(0)=1, u(1)=-1, v(1)=-1.

Now the recurrence relation b(n) = w*b(n-1) + b(n-2) implies that
u(n) = u(n-2) - v(n-1)
v(n) = v(n-2) + u(n-1) - v(n-1)
that in turn implies that
[u(n-1),v(n-1),u(n),v(n)] = M*[u(n-2),v(n-2),u(n-1),v(n-1)]
where vectors are column-vectors and the matrix M is defined as
[ 0, 0, 1, 0 ]
[ 0, 0, 0, 1 ]
[ 1, 0, 0, -1 ]
[ 0, 1, 1, -1 ]

The characteristic polynomial of the matrix M is
x^4 + x^3 - x^2 - x + 1
meaning that u(n) and v(n) satisfies the recurrence
u(n) = - u(n-1) + u(n-2) + u(n-3) - u(n-4)
v(n) = - v(n-1) + v(n-2) + v(n-3) - v(n-4)
with u(0)=0, u(1)=-1, u(2)=1, u(3)=-2
and v(0)=1, v(1)=-1, v(2)=1, v(3)=-1.

>  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.

It is not clear why you don't reduce the "polynomials" of w to the first degree.
Higher degree "polynomials" are not uniquely defined.
E.g., b(1)=w^2=2w^2+w+1=3w^2+2w+2=...
but if you reduce to the first degree "polynomial", there is an unique
representation:
b(1) = -w-1.

[...]

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

This sequence is not well-defined (see above).
Probably, what you are interested in is u(n):
0, -1, 1, -2, 2, -2, 1, 1, -4, 8, -12, 15, -15, 10, 2, -22, 49, -79, 104, -112

Max