  In this paper, we give graphs whose topological index are exactly equal to
the number $u_n$, satisfying the three term recurrence relation $$ u_n=a
u_{n-1}+b u_{n-2}\quad(n\ge 2)\quad u_0=0\quad\hbox{and}\quad u_1=u\,, $$ where
$a$, $b$ and $u$ are positive integers. We show an interpretation from the
continued fraction expansion in a more general case, so that the topological
index can be computed easily. On the contrary, for any given positive integer
$N$, we can find the graphs (trees) whose topological indices are exactly equal
to $N$.
