For a commutative ring with unity $R$, if $R_{red}$ is semi normal then we have an equivalent criterion which states that $Pic(R) \cong Pic(R[t])$. Here $Pic(R)$ is the picard group of $R$.
So naturally it is well understood that $Pic(R) \ncong Pic(R[t])$ when $R = k[t^2,t^3]$ a cusp, which is a domain that is not semi-normal. 

With the help of conductor ideals I was able to show that $Pic(R) \cong k$ but I am unable to explicitly compute $Pic(R[t])$ will the same approach via conductor ideals work, for any suggestion or guidance I am thankful.