Is it true that $$A\_n\equiv (-1)^n\;\;(\mathrm{mod}\;3)\;\;?$$
Here $A\_n$ is the Apery number:
$$A\_n=\sum\limits\_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2.$$
What is known about congruence properties modulo 3 for another set of Apery numbers $$B\_n=\sum\limits\_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}\;?$$



Congruence for the Apery Numbers
Yes, the first congruence was conjectured by Chowla and Cowles in $1980$ and proved by Y. Mimura in $1983$, see <http://www.sciencedirect.com/science/article/pii/0022314X83900380>. For $B\_n$ the sequence begins $(1,3,19,147,1251,11253,104959...)$, which is $(1,0,1,0,0,0,1...)$ modulo $3$. 
It seems we have $B\_{2^k}\equiv 0 \mod 3$ and $B\_n\equiv 1 \mod 3$ otherwise.
Perhaps the recurrence
$$
(n + 1)^2B\_{n + 1} = (11n^2 + 11n + 3)B\_n + n^2B\_{n-1}.
$$
is useful.


Edit: I found a reference here: <http://arxiv.org/abs/math.CO/0407326>. See Theorem $5.8$.


