Eisenstein-Fibonacci sequences

Thu Nov 2 08:44:39 CET 2006

Thank you, Joerg.  That was interesting.  Does your generating function
hypothesis hold when I extend the sequence by another 5 values, to a(n) = 0,
0, 1, 1, 3, 1, 7, 4,
14, 17, 28, 54, 66, 143, 182, 350, 687, 987, 2611, 4298 ? Again, assuming
that I got the elementary algebra and arithmetic correct.

If so, I should submit this sequence as by the two us!

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

[update of 1 Nov 06]

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.

16 w(350 + 300w + 337w^2) + 350 + 300w + 337w^2 =
    350w + 300w^2 + 337w^3 + 350 + 300w + 337w^2 =
    687 +  650w + 637w^2.

17 w(687 +  650w + 637w^2) + 350 + 300w + 337w^2 =
    687w +  650w^2 + 637w^3 + 350 + 300w + 337w^2 =
    987 + 987w + 987w^2.

18  w(987 + 987w + 987w^2) + 687 +  650w + 637w^2 =
    987w + 987w^2 + 987w^3 + 687 +  650w + 637w^2 =
    1674 + 1637w + 1624w^2.

19  w(1674 + 1637w + 1624w^2) + 987 + 987w + 987w^2 =
     1674w + 1637w^2 + 1624w^3 + 987 + 987w + 987w^2
=
     2611 + 2661w + 2624w^2.

20  w(2611 + 2661w + 2624w^2) + 1674 + 1637w + 1624w^2
=
     2611w + 2661w^2 + 2624w^3 + 1674 + 1637w +
1624w^2 =
     4298 + 4248w + 4285w^2.

The coefficients of 1 = a(n) = 0, 0, 1, 1, 3, 1, 7, 4,
14, 17, 28, 54, 66, 143, 182, 350, 687, 987, 2611,
4298

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

Eisenstein-Fib.doc

=== end 2nd email ===

On 11/1/06, Joerg Arndt <arndt at jjj.de> wrote:
>
> * 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]
>
>
> > [...]
>
>
>
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