Thank you to Adrian from South Greenhoe Middle School, Lucy and Sarah from Archbishop Sancroft High School and pupils from Heacham Middle School who all sent correct solutions. Here is a summary of their responses:
The next three numbers in the sequence will be
$1^2 + 2^2 + 3^2 + 4^2 = 30$
$1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55$
$1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91$
The numbers are called pyramid numbers because if you take some balls of the same size and you build them into pyramids, you will need to make square-based pyramids with one ball at the top, four in the next layer down, nine in the next layer down and so on.
Perhaps you can draw pictures for these sequences too?