  Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of
integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray
map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes
over $R$ of odd lengths with the help of cyclic codes over $R$. It is proved
that the Gray image of $(1+2u)$-constacyclic codes of length $n$ over $R$ are
cyclic codes of length $2n$ over $\mathbb{Z}_4$. Further, a number of linear
codes over $\mathbb{Z}_4$ as the images of $(1+2u)$-constacyclic codes over $R$
are obtained.
