On the number of generators of ideals defining Gorenstein Artin algebras with Hilbert function $ (1,n+1, 1+{n+1choose 2},...,{n+1choose 2}+1, n+1,1)$

Let $R = k[w, x_1,..., x_n]/I$ be a graded Gorenstein Artin algebra . Then $I = ann F$ for some $F$ in the divided power algebra $k_{DP}[W, X_1,..., X_n]$. If $RI_2$ is a height one idealgenerated by $n$ quadrics, then $I_2 subset (w)$ after a possible change of variables. Let $J = I cap k[x_1,..., x_n]$. Then $mu(I) le mu(J)+n+1$ and $I$ is said to be generic if $mu(I) = mu(J) + n+1$. In this article we prove necessary conditions, in terms of $F$, for an ideal to be generic. With some extra assumptions on the exponents of terms of $F$, we obtain a characterization for $I = ann F$ to be generic in codimension four.