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

Hint:
Such sums are made of runs of equal values, that are delimited by the indexes such that the general term crosses an integer.
$$\sqrt k+\sqrt{k+n}=m$$ when
$$k+2\sqrt k\sqrt{k+n}+k+n=m^2$$
or
$$4k(k+n)=(m^2-n-2k)^2$$
or
$$k=\frac{(m^2-n)^2}{m^2}=m^2-2n+\frac1{m^2}.$$
As $k$ is an integer, the final fraction can be ignored and a run of weight $m$ has length $(m+1)^2-2n-m^2+2n=2m+1$.
Hence for complete runs, i.e. up to some $k^*=(m+1)^2-2n$ (excluded), the sum is that of $(2m+1)m$.
For the last incomplete run, the length is $n-k^*$, for weight $m+1$.
