TITLE: Are the Gessel sequence integers composite for all $n\ge 3$?
QUESTION [5 upvotes]: The Gessel sequence is known for Ira Gessel's Lattice Path Conjecture of $2001$, which has been proved by Kauers, Koutschan and Zeilberger in $2009$ with the aid of a computer. Later, other proofs were found  ("human proofs"), e.g., by using Weierstrass elliptic functions (see here).
I was wondering whether or not it is true that the integers $a_n$ of the Gessel sequence are never prime numbers for $n\ge 3$. There is an easy recursion for this sequence, namely $a(0):=1$ and 
$$
a_{n+1}=\frac{4(6n+5)(2n+1)}{(3n+5)(n+2)}a_n
$$ 
for all $n$. Is there some (obvious) reason that all $a_n$ are composite ?

REPLY [4 votes]: It is easy to see from the recurrence that every prime factor of $a_{n}$ is less than $6n$.  But the sequence grows much faster than that.