Let $S_n$ be the set of all natural numbers in the decimal system consisting of n digits such that no successive digits of this number is zero.
Let $A_n$ be the number of elements in $S_n$
Then define natural numbers $x,y$ by $$A_9=xA_8+yA_7$$
Find out the square root of $xy$.
The idea of the following argument is the same as the one suggested in lulu's hint, except I chose to start on the left . . .
Fix $n > 2$, and let $s \in S_n$.
Let the digit representation of $s$ be $d_1d_2\cdots d_n$ (left-to-right).
The leading digit of $s$ can be any digit except $0$, hence there are $9$ choices for $d_1$.
Consider two cases . . .
Case $(1)$:$\;d_2 \ne 0$.
Then the $(n-1)$-digit number with digit representation $d_2\cdots d_n$ can be any element of $S_{n-1}$, so for case $(1)$, there are $9A_{n-1}$ possibilities for $s$.
Case $(2)$:$\;d_2 = 0$.
Then the $(n-2)$-digit number with digit representation $d_3\cdots d_n$ can be any element of $S_{n-2}$, so for case $(2)$, there are $9A_{n-2}$ possibilities for $s$.
Summing up the counts for the two cases, we get $A_{n} = 9A_{n-1} + 9A_{n-2}$, for all $n > 2$.
So presumably, $x=9$ and $y=9$ are the intended values of $x,y$.