Maths : finding the general term in a sequential row

[Nineshadow]

We have { x_0, x_1, x_2 } given, and also { A , B , C }  given.

 

And we also have x_3 = A * x_2 + B * x_1 + C * x_0 .

 

More generally : x_n = A * x_n-1 + B * x_n-2 + C * x_n-3 .

 

So now, I need to find out how to calculate x_n without calculating all x-s before it.

 

So far I calculated this:

 

x_4 = x_0 ( AC + B ) + x_1(AB + C) + x_2(A²)

x_5 = x_0(A²C + AB + BC + C) + x_1(A²B + AC + B²) + x_2(A³ + AB)

x_6 =x_0(A³C + A²B + 2ABC + B² + C² + AC) + x_1(A³B + 2AB² + 2BC + A²C) + x_2(A⁴ + 2A²B + AC)

 

Anyone seeing the rule here?Is it even possible?I'm looking for a mathematical way to implement an algorithm. Will be way more efficient this way.

[SuperPug]

Don't you have to solve this with Lagrangian function?

[Nineshadow]

Don't you have to solve this with Lagrangian function?

No idea what that is.

[blackadder]

n!