ELMO SL 2013 N8

We define the Fibonacci sequence $\{F_n\}_{n\ge0}$ by $F_0=0$ , $F_1=1$ , and for $n\ge2$ , $F_n=F_{n-1}+F_{n-2}$ ; we define the Stirling number of the second kind $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets.

For every positive integer $n$ , let $t_n = \sum_{k=1}^{n} S(n,k) F_k$ . Let $p\ge7$ be a prime. Prove that $$t_{n+p^{2p}-1} \equiv t_n \pmod{p}$$ for all $n\ge1$ .

Proposed by Victor Wang