# generalized Binet formula

I google a lot but I'm not able to find any answer. The problem is the following:

For recurrent sequence using the characteristic polynomial it is possible to have a generic formula that gives the $n$-th term in terms of exponential. So for example the Fibonacci sequence $$F_n = F_{n-1} + F_{n-2}; F_0 = 1, F_1 = 1$$ has a Binet formula like

$$F_n = \frac{\phi_n - (-\phi^n)}{\sqrt{5}}$$

Considering a recurrent relation like

$$F_{n,k} = F_{n-1,k} + F_{n-2,k} + \cdots + F_{n-k,k}$$

where all terms from $1$ to $k$ are integers and considering that the characteristic polynomial has a solution, is it possible to have a kind of Binet formula for such a sequence?

Any help would be appreciate

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Yes. It's the same principle but the answer might not be pretty or even expressible in elementary terms. For $k=3$, see wolframalpha.com/input/…. – lhf Jan 9 '12 at 15:06
What's $F(n,k)$? Did you mean just $F_n$? – Hans Lundmark Jan 9 '12 at 16:03