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on When is a cubic-lattice ball Manhattan-optimal?

The statement

For $n\ge 0$ , let $B_n=\{(x,y,z)\in\mathbb{Z}^3:x^2+y^2+z^2\le n\}$ , let $S(n)$ be the sum of Manhattan distances over all unordered pairs in $B_n$ , and let $D(N)$ be the minimum such sum among all $N$ -point subsets of $\mathbb{Z}^3$ . Determine all further $n$ for which $S(n)=D(|B_n|)$ , beyond the reported equality radii $0,1,2,3,6,11,21,30,\ldots$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.