Curious Bounds for Floor Function Sums

The sums of floor functions have been studied by Jacobsthal, 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 the sums and then use it to give proofs of the upper and lower bounds that were claimed by Tverberg. Furthermore, we present conjectural lower and upper bounds for these sums.