An alternating sum of the floor function of square roots

TL;DR

This paper provides a simple evaluation for the alternating sum of the floor of square roots for all odd integers, contrasting with the complexity of non-alternating sums, and includes an asymptotic analysis.

Contribution

It introduces an elementary method, suggested by AI, to evaluate an alternating sum of floor square roots for all odd integers, extending understanding beyond prime-specific cases.

Findings

Explicit evaluation formula for the sum for all odd n

Asymptotic expression for the sum without floor functions

Elementary proof method, aided by AI

Abstract

We show that the alternating sum of the floor function of $\sqrt{jn}$, with $j$ ranging from 1 to $n$, has an easy evaluation for all odd integers $n\geq 1$. This is in contrast to known non-alternating sums of the same type which hold only for a class of primes. The proof is elementary and was suggested by an AI model. To put this result in perspective, we also prove an asymptotic expression for the analogous sum without the floor function.