Odd-distance sets and right-equidistant sequences in the maximum and Manhattan metrics

We solve two related extremal-geometric questions in the $n$-dimensional space $\mathbb{R}^n_\infty$ equipped with the maximum metric. First, we prove that the maximum size of a right-equidistant sequence of points in $\mathbb{R}^n_\infty$ equals 2$^{n+1}$–1. A sequence is right-equidistant if each of the points is at the same distance from all the succeeding points. Second, we prove that the maximum number of points in $\mathbb{R}^n_\infty$ with pairwise odd distances equals 2$^n$. We also obtain partial results for both questions in the $n$-dimensional space $\mathbb{R}^n_1$ with the Manhattan distance.