The [Gessel sequence](https://oeis.org/A135404) 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](https://arxiv.org/abs/1309.1023)).


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 ?



Are the Gessel sequence integers composite for all $n\ge 3$?
Yes. Towards a contradiction, suppose $n \geq 2$ is such that $a\_{n+1}$ is prime. Say, $a\_{n+1}=p$. Then, $(3n+5)(n+2)p=4(6n+5)(2n+1)a\_n$. Evidently, $p$ does not divide $a\_n$ since $a\_{n+1} > a\_n$. Thus, $p$ divides $6n+5$ or $2n+1$. This implies that $6n+5 \geq p=a\_{n+1}$, which is a contradiction for all $n \geq 2$. 


