Number of occurances of a number $n$ in $\lfloor \sqrt{0} \rfloor, \lfloor \sqrt{1} \rfloor, \lfloor \sqrt{2} \rfloor, \dots$

Recently, while learning Python (the programming language), I started playing around to generate the following sequence: $$\lfloor \sqrt{0} \rfloor, \lfloor \sqrt{1} \rfloor, \lfloor \sqrt{2} \rfloor, \lfloor \sqrt{3} \rfloor, \dots$$

Interestingly, this is equivalent to

$$0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, \dots$$

I tried this up to $\lfloor \sqrt{10000} \rfloor$ and saw that every $n \in [0,99]$ occurs exactly $2n+1$ times in the sequence.

Being curious, I'm interested in proving this for all $n \in \Bbb N$ (or finding a counterexample), but I'm not sure how to do this. The only related thing I've found is that determining whether $\lfloor \sqrt{p} \rfloor$ ($p$ is prime) generates all natural numbers is an unsolved problem, but can we do a simple proof of this using all natural numbers? All hints are welcome.

• $\lfloor\sqrt{n}\rfloor = k$ if and only if $k^2 \leqslant n < (k+1)^2$. – Daniel Fischer Apr 27 '14 at 21:28

HINT : All you need to notice is that $$(n+1)^2 - n^2 = 2n+1$$