In a recent paper [ Counting symmetric nilpotent matrices , by A. Brouwer],
the author states that the number of 3x3 symmetric nilpotent
matrices over the field of q elements is given by the formula:
q^3 + q^2 - q
In the case of characteristic 2 , the situation simplifies a
little and one can readily prove the formula.
Question: Is there a straightforward proof in the case of
odd characteristic?