summation - Closed-form for Floor Sum 1

Does a closed form exist for the following sum?
$$\sum_{k=0}^n \lfloor \sqrt{k} + \sqrt{k + n} \rfloor$$

If not, why is this sum so radically different than the sums below?

Closed forms do exist for the following sums*:
$$\sum_{k=0}^n \lfloor \sqrt{k + n} \rfloor$$

$$\sum_{k=0}^n \lfloor \sqrt{k} \rfloor$$

There is this floor functional identity:
$$\lfloor \sqrt{k} + \sqrt{k + 1} \rfloor = \lfloor\sqrt{4k+2}\rfloor$$
Don't know if this will help.

Thanks

*Existing closed forms

$$\sum_{k=0}^n \lfloor \sqrt{k} \rfloor=2\left(\sum_{k=0}^{\lfloor \sqrt{n} \rfloor-1}k^2\right)+\left(\sum_{k=0}^{\lfloor \sqrt{n} \rfloor-1}k\right)+\lfloor\sqrt{n}\rfloor\left(n-\lfloor\sqrt{n}\rfloor^2+1\right)$$
$$\left(\sum_{k=0}^n k^2\right)=\frac{2n^3+3n^2+n}{6}$$
$$\left(\sum_{k=0}^n k\right)=\frac{n^2+n}{2}$$
$$\sum_{k=1}^n \lfloor \sqrt{k+C} \rfloor=\sum_{k=C+1}^{C+n} \lfloor \sqrt{k} \rfloor=\sum_{k=0}^{C+n} \lfloor \sqrt{k} \rfloor-\sum_{k=0}^{C} \lfloor \sqrt{k} \rfloor$$