Suppose that $a_1=3$ and the sequence $(a_n)$ satisfies the recurrence relation

$\frac{a_n-2}{n-1}=\frac{a_{n-1}+2}{(n-1)+1}$
for all $n\ge 2$. What is the greatest integer less than or equal to

${\sum }_{n=1}^{60}{a}_n^2$?