Find the largest possible number of squares 1x1

Question

Martin is playing 210 pieces of 1x1 small squares. What's the largest possible number of squares, all with different side lengths, that he could from using some of the 210 pieces of 1x1 small square?

Answer 1

I can have this, with n=210:

$$ \sum_{i=1}^{\lfloor\sqrt n\rfloor} {n \choose i^2}=\sum_{i=1}^{14} {210 \choose i^2}=7.96806628515312 \cdot 10^{61} $$

The largest term is for $i=10$, ${210 \choose 100}=7.13939343164154\cdot 10^{61}$, which is the closest term for the maximum combinatorial ${210 \choose 105}$.