Integer Subsets with High Volume and Low Perimeter

We explore a variation of the isoperimetric problem in which integer subsets take the role of geometric figures. More specifically, we consider the sequence P (n) introduced by Miller et al. and described in OEIS A186053. We provide the first exact formulas for P (n) including recursive relations via auxiliary functions as well as concise and satisfying representations and even quasi-explicit formulas. We also discuss some of the intricate fractal-like symmetry of the sequence and the development of algorithms for computing P (n). We conclude with open questions for further research.