Curious bounds for floor function sums

Jacobsthal was the first to study a certain kind of sum of floor functions. His work was followed by that of Carlitz, Grimson, and Tverberg. More recently, Onphaeng and Pongsriiam proved some sharp upper and lower bounds for the sums of Jacobsthal and Tverberg. In this paper, we devise concise formulas for these sums, which we then use to prove the upper and lower bounds claimed by Tverberg. Furthermore, we present conjectural lower and upper bounds for these sums.