  We consider a Weitzenb\"ock derivation $\Delta$ acting on a polynomial ring
$R=K[\xi_1,\xi_2,...,\xi_m]$ over a field $K$ of characteristic 0. The
$K$-algebra $R^\Delta = \{h \in R \mid \Delta(h) = 0\}$ is called the algebra
of constants. Nowicki considered the case where the Jordan matrix for $\Delta$
acting on $R_1$, the degree 1 component of $R$, has only Jordan blocks of size
2. He conjectured (\cite{N}) that a certain set generates $R^{\Delta}$ in that
case. Recently Koury (\cite{Kh}), Drensky and Makar-Limanov (\cite{DM}) and
Kuroda (\cite{K}) have given proofs of Nowicki's conjecture. Here we consider
the case where the Jordan matrix for $\Delta$ acting on $R_{1}$ has only Jordan
blocks of size at most 3. Here we use combinatorial methods to give a minimal
set of generators $\mathcal G$ for the algebra of constants $R^{\Delta}$.
Moreover, we show how our proof yields an algorithm to express any $h \in
R^\Delta$ as a polynomial in the elements of $\mathcal G$. In particular, our
solution shows how the classical techniques of polarization and restitution may
be used to augment the techniques of SAGBI bases to construct generating sets
for subalgebras.
