Explanation 15x15 magic square
The 15x15 magic square is odd and 3 x 5. It is possible to us the 3x3 and 5x5 magic square to construct 15x15 composite squares and I show you different options. Use a 5x3 (of 3x5) magic rectangle to construct an ultra magic 15x15 square.
The methods to construct a 15x15 magic square are:
Use the first three methods to construct simple symmetric magic 15x15 squares.
Composite (1) and (2) leads to (simple) 15x15 magic squares consisting of (not proportional) 5x5 respectively 3x3 magic squares.
Composite (3) leads to (simple) 15x15 magic squares with the extra magic feature that each 3x5 or 5x3 rectangle gives the magic sum.
With composite (3a) each 1/5 row/column of the 15x15 magic square gives 1/5 of the magic sum and the 15x15 magic square is 3x3 compact. Alternative is symmetric instead of 3x3 compact.
With composite (4) you can construct a 15x15 magic square consisting of 9 proportional 5x5 panmagic squares. This 15x15 magic square is panmagic, 5x5 compact (but not symmetric) and each 1/3 row/column/diagonal gives 1/3 of the magic sum.
Use the shift methods to construct panmagic 15x15 squares (shift method (2) gives a more tight structure).
It is also possible to use 9x a 5x5 panmagic square and 2 ternary grids to construct a panmagic and 5x5 compact 15x15 magic square.
Use a 5x3 (or 3x5) magic rectangle to construct a 15x15 ultra magic square, which is panmagic, symmetric, 3x3 compact and 5x5 compact.
Finally construct a 3x3 in 5x5 in 7x7 in 9x9 in 11x11 in 13x13 in 15x15 (concentric) magic square.