Eisenstein-Fibonacci sequences

[Max A.]

On 11/9/06, Jonathan Post <jvospost3 at gmail.com> wrote:

> Max's position is well taken.  I think that "random" overstates it somewhat.

OK, it may not be random from your point of view since you followed
some rules (I believe) defining the terms of this sequence. The
problem is that you did not let us know these rules!
Remember the example I gave above? It was about different
representations of b(1) as 2-nd degree polynomials of w:
b(1) = w^2 = 2w^2+w+1 = 3w^2+2w+2 = ...
They all are equal (since w=(-1 + i*sqrt(3))/2) but at the same time
they all have different coefficients as polynomials of w.
For some (unknown to us) reason, you chose the first representation
that gave the 1st term of your sequence equal 0 (as the coefficient of
w^0 in b(1)=w^2). But you chose b(1)=2w^2+w+1, the first term would be
1; if you chose b(1)=3w^2+2w+2, it would be 2, etc.

There is yet another way to define your sequence precisely. This time
let's forget the exact value of w and keep in mind only that w^3 = 1.
Then every element of the sequence b(0) = w, b(1) = w^2, b(n) =
w*b(n-1) + b(n-2) have unique representation as a 2-nd degree
polynomial of w:
b(2) = -x + 1
b(3) = -2*x^2 + x
b(4) = x^2 + x - 3
b(5) = 3*x^2 - 4*x + 1
b(6) = -5*x^2 + 6
b(7) = -3*x^2 + 10*x - 6
b(8) = 15*x^2 - 6*x - 9
...
This gives us the following sequence of coefficients of w^0:
0, 0, 1, 0, -3, 1, 6, -6, -9
But it is different from your original sequence (again).

Max