  We study the period of the linear map $T:\mathbb{Z}_m^n\rightarrow
\mathbb{Z}_m^n:(a_0,\dots,a_{n-1})\mapsto(a_0+a_1,\dots,a_{n-1}+a_0)$ as a
function of $m$ and $n$, where $\mathbb{Z}_m$ stands for the ring of integers
modulo $m$. Since this map is a variant of the Ducci sequence, several known
results are adapted in the context of $T$. The main theorem of this paper
states that the period modulo $m$ can be deduced from the prime factorization
of $m$ and the periods of its prime factors. We also characterize the tuples
that belong to a cycle when $m$ is prime.
