I'm studying Elliptic Curves and EDS (Elliptic divisibility sequences) and working on Silvermans exercises 3.34 in "The arithmetic of elliptic curves":
"An EDS over $K$ is a Sequence $(W_n)_{n\geq 1}$ defined by four initial conditions $W_1, W_2, W_3, W_4 \in K$ ans satisfying the recurrence:
$W_{m+n}W_{m-n}W_1^2=W_{m+1}W_{m-1}W_n^2-W_{n+1}W_{n-1}W_m^2$
I already proved that the even terms of the Fibonacci-sequence satisfy this recurrence.
The next exercise is to generalize this result and find a subsequence of the sequence $(L_n)_{n\geq 1}$:
$L_1=1 \quad L_2=P \qquad \qquad \qquad L_{n+2}= PL_{n+1} - L_n$
which also saitsfies the recurrence above.
I already recognized the the given formula describes a Lucas-sequence : $L_n= \frac{a^n-b^n}{a-b} $ with $P=a+b, Q=ab=1$
I just need an idea to proof that such a sequence satisfies the recurrence relation above.