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Does a closed form exist for the following sum?

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

where $P$ is a positive odd integer. Would a closed-form be possible if the factorization of $P$ is known?

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  • $\begingroup$ You have $n$ and $N$ here - are they meant to be the same or different values? $\endgroup$ – Steven Stadnicki Aug 28 '17 at 19:06
  • $\begingroup$ They are different, $N$ in this case is a positive odd integer. $\endgroup$ – Ruan Sunkel Aug 28 '17 at 19:07
  • $\begingroup$ I would suggest not using such similar variable names in general; confusion is almost inevitable. In any case, a closed form seems highly unlikely particularly given the 'freeform' nature of $N$ - transitions in the sum can happen at essentially any point and tracking them looks highly difficult. $\endgroup$ – Steven Stadnicki Aug 28 '17 at 19:10
  • $\begingroup$ The problem is, however, interesting for $N=n$. $\endgroup$ – Mark Fischler Aug 28 '17 at 19:13
  • $\begingroup$ Good advice, I will update the variable names $\endgroup$ – Ruan Sunkel Aug 28 '17 at 19:15

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