proof outline for square pyramidal numbers 
It is possible to write down this formula:

This is an alternative way to group the sum of the squares:
to quote an example, we assume n=5:

In this representation the number five is summed 5 times,
the number 4 is summed 4 times, and so on, until zero.
After that, it is possible to use this result to solve the summation:

The main formula is equivalent to

After some agebraic manipulation it is possible to obtain:

This is equivalent to the desired formula.