Maurice Kraitchik gave the number of magic series up to n = 7 in Mathematical Recreations in 1942 (sequence A052456 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.
Trump used this new approach to extend the sequence up to n = 1000.
[1] Mike Quist showed that the exact second-order count has a multiplicative factor of
[2] Richard Schroeppel in 1973 published the complete enumeration of the order 5 magic squares at 275,305,224.
It might be described as an amalgamation of 4 magic series that have only one unique common integer.
This structure forms the two major diagonals and the central row and column for an odd order magic square.