I do not understand how to obtain the stated formula for the characteristic path length (L) of a $1$-lattice, as well as its clustering coefficient. The following text is taken from The Small World Phenomenon by Duncan Watts.
Definition The characteristic path length (L) of a graph is the median of the means of the shortest path lengths connecting each vertex $v\in V(G)$ to all other vertices. That is, calculate $d(v,j),\forall j\in V(G)$ and find $\overline{d}_v$ for each $v$. Then define $L$ as the median of $\{\overline{d}_v\}$.
A $d$-lattice is a labelled, unweighted, undirected simple graph that is similar to a Euclidean cubic lattice of dimension $d$ in that any vertex $v$ is joined to its lattice neighbours $u_i$ and $w_i$ are specified by \begin{eqnarray} u_i&=&\left[\left(v-i^{d^{'}} \right)+n\right] \mod n \\ w_i&=&\left( v+i^{d^{'}}\right) \mod n \end{eqnarray}
where $1\le i\le \frac{k}{2}, 1\le d^{'}\le d$ and $k\ge 2d$. Below is an illustration of a $1$-lattice with $k=4$.
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For a $1$-lattice with even $k\ge 2$, the characteristic path length is $$ L = \frac{n(n+k-2)}{2k(n-1)} $$
Its clustering coefficient is $$ \gamma =\frac{3}{4}\left( \frac{k-2}{k-1}\right) $$
From what I can gather, since each vertex is isomorphic in a one lattice, L is just the mean of shortest path lengths between a vertex v and any other vertex.
The (n-1) part of the denominator comes from the fact that this is a mean of paths between n-1 vertices. Number these other vertices one through n-1. Their summed distance will be: $\sum_{i=1}^{n-1} ceiling(\frac{min(i,n-i)}{k/2})$, the numerator represents the distance of this was just a cycle, not a 1-lattice, and the k represents the maximum cycle distance you may travel. Intuitively, the numerator should be ~n(n-1)/4, consequence of its resemblance of an arithmetic sum, meanwhile the denominator is k/2.
$\frac{n(n-1)/4}{k/2} = \frac{n(n-1)}{2k}$
This largely accounts for the terms in the characteristic path length. The +n(k-1)/2k should arise from cleaning up details like the ceiling function.