Need some help from Fibonacci experts...

[Ralf Stephan]

> I have some expressions that I simplified as much as I could, but
> I wonder if someone can find simpler. Thank you very much.
> 
> I have to tell that n is even in all those expressions, which may
> allow some further simplifications : (-1)^n = 1
> 
> 		       2	       2
> (%o759) - 20 FIB(n) FIB (n + 1) - 4 FIB (n) FIB(n + 1) - 12 FIB(n + 1)

Go generating functions. This has ogf:

(4*x^4 + 4*x^3 + 80*x^2 + 12*x - 12)/(-x^6 - 4*x^5 + 4*x^4 + 12*x^3 - 4*x^2 - 4*x + 1)

Factoring the denominator yields

(1-4x-x^2) * (1-x-x^2) * (1+x-x^2)

so the expression must be a sum of three functions that have ogf
with denominator being each of the factors.
In fact, any sequence in the OEIS with such an ogf will work, e.g.,

Expr = A*F(3n+i) + B*F(n+j) + C*(-1)^(n+k)*F(n+k).

This solution space can then be searched with LLL, one of the most
useful algorithms we have.
If you have such problems often, you would be wise to develop a tool
for yourself now. I could solve Expr for you tomorrow so you might
want to wait but I offer help in implementing, too. Which one do
you want?


ralf
PS: Or maybe even my hints suffice?