HMMT 2016 U33

(Lucas Numbers) The Lucas numbers are defined by $L_{0}=2, L_{1}=1$, and $L_{n+2}=L_{n+1}+L_{n}$ for every $n \geq 0$. There are $N$ integers $1 \leq n \leq 2016$ such that $L_{n}$ contains the digit 1 . Estimate $N$. An estimate of $E$ earns $\lfloor 20-2|N-E|\rfloor$ or 0 points, whichever is greater.