  Let $K$ be a field and $X$, $Y$ denote matrices such that, the entries of $X$
are either indeterminates over $K$ or $0$ and the entries of $Y$ are
indeterminates over $K$ which are different from those appearing in $X$. We
consider ideals of the form $I_{1}(XY)$, which is the ideal generated by the
homogeneous polynomials of degree $2$ given by the $1\times 1$ minors of the
matrix $XY$. We prove that $d$-sequences and regular sequences arise naturally
as part of generators of $I_{1}(XY)$ for some special cases. We use this
information to calculate the equations defining the Rees algebra of
$I_{1}(XY)$.
