Suppose I have a quadratic equation of the form $x^2-2bx-c=0$ with roots
$k_1=b+\sqrt{b^2+c}$ , and
$k_2=b-\sqrt{b^2+c}$
Is there a general formula for
$t_n=(k_1)^n-(k_2)^n$?
I wanted to calculate this because I wanted to know, given $t_n$ and $t_{n+1}$ can we calculate $(b,c)$?
Edit: changed mistake in formula for $k_1,k_2$