Ducci on ${\mathbb{Z}}_m^n$ and the Maximum Length for $n$ Odd

Define the Ducci function $D:{\mathbb{Z}}_m^n\to {\mathbb{Z}}_m^n$ so $$D(x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_n)=(x_1+x_2\text{mod}m,x_2+x_3\text{mod}m,\mathrm{.}\mathrm{.}\mathrm{.},x_n+x_1\text{mod}m)\mathrm{.}$$ Call $\{D^{\alpha }(\mathbf{u}){\}}_{\alpha =0}^{\mathrm{\infty }}$ the Ducci sequence of $\mathbf{u}$. Because ${\mathbb{Z}}_m^n$ is finite, every Ducci sequence will enter a cycle. In this paper, we will prove that if $n$ is odd and $m=2^l{m}_1$ where $m_1$ is odd, then the longest it will take for a Ducci sequence to enter its cycle is $l$ iterations. Furthermore, we will prove the set of all tuples in a cycle for ${\mathbb{Z}}_m^n$ is $\{(x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_n)\in {\mathbb{Z}}_m^n\mid x_1+x_2+\cdots +x_n\equiv 0\text{mod}{2}^l\}$.