Closed form for recursive sequence mod p When we have recursive sequences, we often seek to define them in a closed form if possible. Yet sometimes, these recursive sequences don't have closed forms. So my question is, is there any recursively defined sequence which doesn't have a closed form, but does have a closed form mod p? ie, ($a_n$) doesn't have a closed form, but ($a_n(\mod p)$) does?

It is really much harder than you think to nail down precisely what "closed form" means. Your definition in the comments is really not sufficient: for example, does "$a_n = 1$ if $n$ is prime and $0$ otherwise" count as a closed form?
Anyway, assuming you're only asking about one prime, the answer is yes for dumb reasons. Consider, for example, the recurrence
$$a_n = p a_{n-1}^3 + 1, a_0 = 1.$$
As far as I know, this doesn't have a closed form in any reasonable sense, but $\bmod p$ the recurrence reduces to $a_n = 1$, so this sequence is constant $\bmod p$. 
