Find a square square pyramidal number

The metapuzzle involves explaining why the word "square" is repeated.

"Find a round square number" is easy enough. An answer is 100.
"Find a round triangular number" is also easy. An answer is 10.
"Find a round square number" has uninteresting answers.

A square pyramidal number is one such that you can build a pyramid from that many cannonballs, with n² on the ground in a square, (n-1)² resting on top of them, and so on.

A square square pyramidal number is one which is both square and square pyramidal.

Restated puzzle: show

Solution to puzzle: show

Apart from 0 and 1, which may be considred trivial, the only solution is 4900, which is the sum of the first 24 positive squares. (This solution is asserted here on proofwiki to be unique, without a proof.)

Comments: show

I hope the metapuzzle is not too hard. The solution can be found by an obvious but tedious search.

I was surprised to find that this is a well-documented puzzle, known as the cannonball problem; and that its solution is related to the Leech lattice.

Find a square square pyramidal number

Problem statement

Solution to metapuzzle: show

Restated puzzle: show

Solution to puzzle: show

Comments: show