Magic series

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A magic series is a set of distinct positive numbers which add up to the magic sum of a magic square, thus potentially making up a line in a magic square.

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, and there is no magic square. 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 A052456 in 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:

\frac{1}{\pi} \cdot \sqrt{\frac{3}{e}} \cdot \frac{(e n)^n}{n^3-\frac{3}{5}n^2+\frac{2}{7}n}

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

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