If we write down the floor values we see
$$
S = 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 4 + ... + 44
$$
(note how, conveniently, $$\sqrt{2024}$$ is just at the border between 44 and 45)
or
$$
S = \sum_{k=1}^{44}{k \times (2k + 1)}
$$
or
$$
S = 2 \times \sum_{k=1}^{44}{k^2} + \sum_{k=1}^{44}{k}
$$
and then using the Sums of powers formula from the suggested page
$$
S = 2 \times \frac{44^3}{3} + \frac{44^2}{2} + \frac{44}{6} + \frac{44*45}{2} = 59730
$$


Scroogey

science

[Daily puzzle] Sum of floors

south_korea_ln

[Empirically](https://glot.io/snippets/h1m2xvq4bq) $$S=59730$$.