A linear recurrent sequence is a recurrent sequence that, for some natural
* n,* all the terms of the sequence starting at * n+1 * are
a fixed linear combination (with coefficients in the complex field) of the
last * n * terms.

One of the things I made is that, given a collection of complex numbers, to find
a linear recurrent sequence of minimum order (order is what * n *
represents in the last paragraph) which starts with the given numbers.
For this purpose I've used some linear algebra objects.

Another thing was to find a majorant and a minorant for the order of the sum of two linear sequences, and to determine one case in wich that majorant is reached.

The main theory that I used is in 1 and here theory.dvi.gz(DVI format), theory.ps.gz (PostScript format) are some corollaries that I needed.

The implementation is made in Squeak , click here to download the code.

- Linear recurrent sequences
- The sequence of partial sums of a linear recurrent it's also
linear recurrent, and a majorant for the order is k+1 (where k is the order
of the first one).
Another nice thing would be: given a linear recurrent

*a*determine if exists another*b*that verifies "*a*is the sequence of partial sums of*b*" - Ask for the closed formula of a given sequence (one way to do this is
implementing Jordan's form, I've already implement it so if you want it just
mail me).
- Define a multiplicative operation:
The convolution (Cauchy's product) and term by term product are also linear reccurrent sequence operations (I don't have the proof of this, not even a majorant for the orders).

- The sequence of partial sums of a linear recurrent it's also
linear recurrent, and a majorant for the order is k+1 (where k is the order
of the first one).
- Build another kind of recurrent sequences
- Sequences with coefficients in K[X].
- Sequences with coefficients in the field of rational functions of K[X].
- Anything else that comes to your mind (the two last items are pretty interesting).

Eric Rodriguez Guevara eguevara@dc.uba.ar

[1] Juan Sabia ( jsabia@dm.uba.ar ) , Susana Tesauri ( stesauri@dm.uba.ar ) , Sucesiones Recursivas Lineales,Notas de Matematica,5(U.B.A. 1997) recursivas.dvi.gz (DVI format) recursivas.ps.gz (PostScript format)