[seqfan] A bijection involving 3x3 magic squares

[David Radcliffe]

Sequence A059329 is described as the number of 3 X 3 matrices, with
elements from {0,...,n}, having the property that the middle element of
each of the eight 3-element horizontal, vertical and diagonal lines equals
the average of the two end elements.

I found this sequence while attempting to solve a different problem: What
is the number of 3 X 3 magic squares with elements from {0, ..., n}, with
repetitions allowed?

I was surprised that these questions seem to have the same answer. After a
bit of thought, I found that there is a bijection between the two types of
arrays. If
A B C
D E F
G H I
is a 3 X 3 matrix satisfying the definition of A059329, then
H A F
C E G
D I B
is a 3 X 3 magic square, and conversely.

For example,
0 1 2
3 4 5
6 7 8
is a 3 X 3 matrix satisfying the condition of A059329, and the
corresponding 3 X 3 magic square is
7 0 5
2 4 6
3 8 1

- David Radcliffe