Asymptotic enumeration of magic series

TL;DR

This paper derives an integral representation and asymptotic expansion for counting magic series, providing accurate approximations and extending results to higher-dimensional magic configurations.

Contribution

It introduces a novel integral representation and asymptotic expansion for the enumeration of magic series, including higher-dimensional cases.

Findings

Excellent agreement between approximation and exact values

Derived asymptotic formulas for magic cube and hypercube series

Extended methods to bimagic and trimagic series

Abstract

A magic series is a set of natural numbers that, by virtue of its size, sum, and maximum value, could fill a row of a normal magic square. In this paper, we derive an exact two-dimensional integral representation for the number of magic series of order N. By applying the stationary phase approximation, we develop an expansion in powers of 1/N for the number of magic series and calculate the first few terms. We find excellent agreement between our approximation and the known exact values. Related results are presented for magic cube and hypercube series, bimagic series, and trimagic series.