Magic series

A magic series is a set of distinct positive numbers which add up to the magic constant of a magic square and a magic cube, thus potentially making up lines in magic tesseracts.

So, in an n × n magic square using the numbers from 1 to n², a magic series is a set of n distinct numbers adding up to n(n²+1)/2. For n = 2, there are just two magic series, 1+4 and 2+3. The eight magic series when n = 3 all appear in the rows, columns and diagonals of a 3 × 3 magic square.

Maurice Kraitchik gave the number of magic series up to n = 7 in Mathematical Recreations in 1942 (sequence in the OEIS). In 2002, Henry Bottomley extended this up to n = 36 and independently Walter Trump up to n = 32. In 2005, Trump extended this to n = 54 (over 2×10¹¹¹) while Bottomley gave an experimental approximation for the numbers of magic series:

In July 2006, Robert Gerbicz extended this sequence up to n = 150.

In 2013 Dirk Kinnaes was able to exploit his insight that the magic series could be related to the volume of a polytope. Walter Trump used this new approach to extend the sequence up to n = 1000.

Mike Quist showed that the exact second-order count has a multiplicative factor of ${\displaystyle {\tfrac {1}{n^{3}}}\left(1+{\tfrac {3}{5n}}+{\tfrac {31}{420n^{2}}}+\cdots \right)}$ equivalent to a denominator of ${\displaystyle n^{3}-{\tfrac {3}{5}}n^{2}+\left({\tfrac {2}{7}}+{\tfrac {1}{2100}}\right)n+\cdots .}$

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