  K. Konno proved a slope equality for fibred surfaces with fibres of odd genus
and general fibre of maximal gonality. More precisely he found a relation
between the invariants of the fibration and certain weights of special fibres
(called the Horikawa numbers). We give an alternative and more geometric proof
in the case of a genus 5 fibration, under generality assumptions. In our
setting we are able to prove that the fibre with positive Horikawa numbers are
precisely the trigonal ones, we compute their weights explicitly and thus we
exhibit explicit examples of regular surfaces with assigned invariants and
Horikawa numbers.
