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There are 7 colours for rotational symmetry order 4, 5 by 5, as there are 6 colours in each corner, being repeated 4 times, and the one centre colour. 6+1=7

There are 13 colours for rotational symmetry order 4, 7 by 7, as there 12 colours in each corner, being repeated 4 times, and the one centre colour. 12+1=13

The formula in terms of n, when n is an odd number of rows is,
(n2-1)/4 +1

This works as the n2-1 is the whole pattern, minus the centre colour, which can be divided in to four parts. EG the 6 squares in the corner in the 5 by 5 pattern.

The dividing by four is being specific to one of the corner colours, which is repeated four times, and the plus one is adding the centre colour we had to subtract to divide by four.