Here is a nice identity which according to  appeared in a 1957 Chinese mathematics competition.
An elegant proof of this avoids any lengthy expansions. Let , and be roots of the cubic polynomial
and summing the relation for each of and , gives
In a similar manner, and so
Next, and so
Finally, and so
We then combine (3), (6) and (7) to obtain (1). It seems that for higher values of are more complicated expressions in and latex $b$, so we don’t get as pretty a relation elsewhere.
 Răzvan Gelca and Titu Andreescu, Putnam and Beyond, Springer, 2007.