Anti-Magic Squares

Today I turned 25307 days old and, as is my habit, I examined the entries for this number in the Online Encyclopaedia of Integer Sequences or OEIS. I've already mentioned 25307 in my previous post as forming part of a prime quadruple consisting of 25301, 25303, 25307 and 25309. For this reason, it is mentioned in OEIS A136721 Prime quadruples: 3rd term.

However, I also came across a mention of the number in OEIS A117560 $n(n^2-1)/2 - 1$ which at first sight seemed unremarkable. However, on reading further I discovered that the numbers in this sequence form an:

approximation for the lower bound of the "anti-magic constant" of an anti-magic square of order n. The anti-magic constant here is defined as the least integer in the set of consecutive integers to which the rows, columns and diagonals of the square sum. 

I was of course familiar with magic squares but I'd never heard of anti-magic squares. I was prompted to investigate further.

To quote from WolframAlpha:

An anti-magic square is an $n×n$ array of integers from $1$ to $n^2$ such that each row, column, and main diagonal produces a different sum such that these sums form a sequence of consecutive integers. It is therefore a special case of a heterosquare. It was defined by Lindon (1962) and appeared in Madachy's collection of puzzles (Madachy 1979, p. 103), originally published in 1966.

There are no $2 \times 2$ or $3 \times 3$ anti-magic squares. The first occurrence is of a $4 \times 4$ anti-magic square and shown below are examples of anti-magic squares from $4 \times 4$ to $9 \times 9$:

Source: http://www.magic-squares.net/anti-ms.htm

The formula $0.5 \times n(n^2-1)-1$ yields 251 in the case of $n=8$ but it should be remembered that this number is a lower bound. There are $8 \times 8$ anti-magic squares that start with a higher number. Below is a diagram of an $8 \times 8$ anti-magic square showing the various totals from 252 to 269 (as opposed to 251 to 268 in the example above):

In the case of 25307, the value of $n$ is 37. So the sum of any sequences formed by adding the elements in any row, column or diagonal of this $37 \times 37$ anti-magic square of numbers have 25307 as their lower bound.

There is a similar sequence for the upper bounds of anti-magic squares. The formula for generating the terms is: $$\lfloor \frac{n(n^3-n-3)}{2(n-1)} \rfloor$$ The OEIS is A117561.