Sets with High Volume and Low Perimeter

TL;DR

This paper investigates a variation of the isoperimetric problem for subsets of nonnegative integers, providing exact formulas, recursive relations, and algorithms for the sequence P(n), revealing its fractal-like symmetry.

Contribution

It introduces the first exact formulas and recursive relations for the sequence P(n) related to the isoperimetric problem on integers, along with new algorithms for computation.

Findings

Derived exact formulas for P(n)

Established recursive relations involving auxiliary functions

Developed algorithms for efficient computation

Abstract

In this paper, we consider a certain variation of the "isoperimetric problem" adopted for subsets of nonnegative integers. More specifically, we explore the sequence P(n) as described in OEIS A186053. We provide the first exact formulas for P(n) including multiple recursive relations involving auxiliary functions as well as concise and satisfying representations and quasi-explicit formulas. We also discuss some of the intricate fractal-like symmetry of the sequence as well as the development of algorithms for computing P(n). We conclude with open questions for further research.