Combining two second-order linear recurrences

[Kieren]

Hello all,

I'm trying to solve a system of two simultaneous Diophantine
equations. I have found two second-order linear recurrence equations,

A(n) = rA(n-1) + sA(n-2)
B(n) = uB(n-1) + vB(n-2).

I have determined that A(0) = B(0) = 0 and A(2) = B(1), corresponding
to the first two solutions of the original system. Is there a way to
"combine" such recurrences, i.e., can I derive a third recurrence

C(n) = pC(n-1) + qC(n-2) + ...

[of preferably second-order, but any order would do for now], which
would show *all* intersections of A(a) & B(b) for a,b > 2?

Thanks,
Kieren.

[Mariano Suárez-Alvarez]

What do you mean by "intersections"?

-- m