An identity that comes from computing the Wiener index of a cyclic graph Can the below identity be proven in such a way that we can generalize it?
$(1 + 1 + 2 + 2 + 3 + 3 + 4) +( 1 + 2 + 2 + 3 + 3 + 4) + (1 + 2 + 3 + 3 + 4)+ +( 1 + 2 + 3 + 4 )+(1 + 2 + 3) + (1 + 2) + 1 = {4^3}$
This comes from computing the Wiener index for the cyclic graph with $8$ vertices.

Note that you can calculate the Weiner index of the 2n-gon as
$\dfrac 12\cdot 2n\left(2\cdot1+2\cdot2+\cdots+2\cdot (n-1)+n\right)$
