Where can I find a list of similar algebra formulas? Here is a simple formula:
$$(n - 1)\sum_{i = 1}^n x_i^2 - 2\sum_{i = 1}^n\sum_{j = 1}^n x_ix_j = \sum_{i = 1}^n\sum_{j = 1}^n(x_i - x_j)^{2}.$$
As a self-learner, I want to know what is the English word to describe this kind of formula? I tried to google "series formula", but seems not a proper word.
So what would be the proper word for this kind of formulas, so that I can get a list of them? 
Thanks a lot.

What you have written is a "summation identity" (or, less precisely, a "summation formula"). You can find many lists of summation identities with a bit of internet searching.
