We first define a new Gray map from R=Z4+uZ4 to , where u2=1 and study (1+2u)-constacyclic codes over R . Also of interest are some properties of (1+2u)-constacyclic codes over R . Considering their Z4 images, we prove that the Gray images of (1+2u)-constacyclic codes of length n over R are cyclic codes of length 2n over Z4. In many cases the latter codes have better parameters than those in the online database of Aydin and Asamov. We also give a corrected version of a table of new cyclic R-codes published by Özen et al.