Numerical evidence for the first hundred Apery numbers
$$A\_n=\sum\limits\_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2$$
suggests the following congruence relation
$$A\_n\equiv 0\; (\mathrm{mod}\; 5),\;\;\mathrm{if}\;\; n\equiv \{1,3\}\;(\mathrm{mod}\; 5).$$
Was this congruence ever proved?



Conjectured congruence for the Apery numbers
The Apery numbers satisfy the recurrence 
$$ 
n^3 A\_n = (34n^3-51n^2+27n-5)A\_{n-1}- (n-1)^3 A\_{n-2}.
$$ 
If $n\equiv 1\pmod 5$ this recurrence gives 
$$ 
A\_n \equiv (4\cdot 1-1\cdot 1+2\cdot 1-5)A\_{n-1} \equiv 0 \pmod 5.
$$ 
If $n\equiv 3 \pmod 5$ it gives 
$$ 
2 A\_n \equiv (4 \cdot 2-1\cdot 4+2\cdot 3 -5)A\_{n-1}-3A\_{n-2} \equiv 2 A\_{n-2} \equiv 0 \pmod 5,
$$ 
by the result just established.


P.S. This proof was also given in the paper S. Chowla, J. Cowles, M. Cowles, Congruence properties of Apéry numbers J. Number Theory, 12 (1980), pp. 188–190, as indicated by Pietro Majer.


