Sequence of Fibonacci Numbers ending in Index

Theorem
Let $F_k$ denote the $k$th Fibonacci number.

For all $k \in \Z$, let $F_k$ be expressed in decimal notation.

The sequence of integers $\left\langle{n}\right\rangle$ such that $F_n$ ends in $n$ starts:
 * $0, 1, 5, 25, 29, 41, 49, \ldots$