Solution

25555

First name
Joseph Lee
School
Wilsons Grammer School
Country
Age
14

There 9 colours for rotational symmetry order 2 and 2 lines of symmetry, 5 by 5, as there as 4 colours in the corner, repeated 4 times, 2 colours on either centre column, repeated 2 times, and the centre colour. 4+2+2+1=9.

There 16 colours for rotational symmetry order 2 and 2 lines of symmetry, 7 by 7, as there as 9 colours in the corner, repeated 4 times, 3 colours on either centre column, repeated 2 times, and the centre colour. 9+3+3+1=16.

The formula in terms of n, when n is an odd number of rows is,
(n2-1)/4 + (n-1)/2 + 1 (used for explanation) or (n2+2n+1)/4
(simplified).

This works as n2-1 is the square, minus the centre, giving you the rectangle of different colours at the top left, also repeated at bottom right (rectangle width facing left or right). The (n-1)/2 are the colours on the vertical centre column and the last +1 is the centre colour we subtracted to divide by four.