Re^2: Eisenstein-Fibonacci sequences

[Richard Mathar]

ma> From seqfan-owner at ext.jussieu.fr  Thu Nov  2 13:12:31 2006
ma> Return-Path: <seqfan-owner at ext.jussieu.fr>
ma> Date: Thu, 2 Nov 2006 04:06:51 -0800
ma> From: "Max A." <maxale at gmail.com>
ma> To: "Jonathan Post" <jvospost3 at gmail.com>
ma> Subject: Re: Eisenstein-Fibonacci sequences
ma> Cc: "Sequence Fans Mailing List" <seqfan at ext.jussieu.fr>
ma> 
ma> On 11/2/06, Max A. <maxale at gmail.com> wrote:
ma> 
ma> > >  Let b(0) = w, b(1) = w^2, b(n) = w*b(n-1) + b(n-2),
ma> > >  where w = omega = (-1 + i*sqrt(3))/2 and w^2 = omega^2
ma> > >  = (-1 - i*sqrt(3))/2.
ma> >
ma> > First off, using the identity w^2 = -w-1, we can uniquely represent
ma> > b(n) as u(n)+v(n)*w where u(n) and v(n) are integers.
ma> 
ma> [...]
ma> 
ma> > u(n) and v(n) satisfies the recurrence
ma> > u(n) = - u(n-1) + u(n-2) + u(n-3) - u(n-4)
ma> > v(n) = - v(n-1) + v(n-2) + v(n-3) - v(n-4)
ma> > with u(0)=0, u(1)=-1, u(2)=1, u(3)=-2
ma> > and v(0)=1, v(1)=-1, v(2)=1, v(3)=-1.
ma> 
ma> o.g.f. for u(n) is
ma> -x/(1 + x - x^2 - x^3 + x^4)
ma> and
ma> o.g.f. for v(n) is
ma> (1-x^2)/(1 + x - x^2 - x^3 + x^4)
ma> 
ma> Hence, o.g.f. for b(n) is
ma> (-x + w*(1-x^2)) / (1 + x - x^2 - x^3 + x^4)
ma> 
ma> Max

The g.f. above [-x+w(1-x^2)]/[1+x-x^2-x^3+x^4] generates
the reduced  polynomial
                          2             3      4            5               6
w + (-1 - w) x + (1 + w) x  + (-2 - w) x  + 2 x  + (w - 2) x  + (-3 w + 1) x

                  7               8              9      10
     + (1 + 5 w) x  + (-4 - 7 w) x  + (8 + 8 w) x  + O(x  )

If one doesn't reduce the polynomial, one gets the simpler
g(x)=w/[1-w*x-x^2] which generates the non-reduced and more generic
b(0)+b(1) x+ b(2) x^2 + b(3) x^3+ ...
=

     2       3       2     4      2   3     5      3       4
w + w  x + (w  + w) x  + (w  + 2 w ) x  + (w  + 3 w  + w) x  +

      6      4      2   5     7      5      3       6
    (w  + 4 w  + 3 w ) x  + (w  + 5 w  + 6 w  + w) x  +

      8      6       4      2   7     9      7       5       3       8
    (w  + 6 w  + 10 w  + 4 w ) x  + (w  + 7 w  + 15 w  + 10 w  + w) x  +

      10      8       6       4      2   9      10
    (w   + 8 w  + 21 w  + 20 w  + 5 w ) x  + O(x  )

and which becomes (apart from an offset) A000045 for w=1, A052542 for w=2,
A052906 for w=3, A000007 for w=0, and signed variations of these for w=-1,-2 or -3..
The coefficients table of the non-reduced g.f. is in A102426, which is
essentially the same as A098925 and A092865.

R. Mathar