# How to find a short form of recursive defined sequences?

Hi, I'm new to sagemath.

Is there any way to so calculate/solve/find a short version of a recursive defined sequence?

E.g. I have a sequence like: (Fibonacci)

```
def f(n):
if n == 0:
return 0
if n == 1:
return 1
if n == 2:
return 1
else:
return f(n-1)+f(n-2)
```

How can I compute a short form of $f_n$?

In this example case $f_n$ would be:

$f_n=\frac{1}{\sqrt{5}} (\frac{1+\sqrt{5}}{2})^n - \frac{1}{\sqrt{5}} (\frac{1-\sqrt{5}}{2})^n$

Edit: Thanks to Emmanuel I found how to solve those equations in pdf:

```
from sympy import Function,rsolve
from sympy.abc import n
u = Function('u')
f = u(n-1)+u(n-2)-u(n)
rsolve(f, u(n), {u(0):0,u(1):1})
-sqrt(5)*(1/2 - sqrt(5)/2)**n/5 + sqrt(5)*(1/2 + sqrt(5)/2)**n/5
```

Now read the book, cover to cover, and keep it under your pillow...