  The Somos 4 sequences are a family of sequences satisfying a fourth order
bilinear recurrence relation. In recent work, one of us has proved that the
general term in such sequences can be expressed in terms of the Weierstrass
sigma function for an associated elliptic curve. Here we derive the analogous
family of sequences associated with an hyperelliptic curve of genus two defined
by the affine model $y^2=4x^5+c_4 x^4+ ... +c_1 x+c_0$. We show that the
recurrence sequences associated with such curves satisfy bilinear recurrences
of order 8. The proof requires an addition formula which involves the genus two
Kleinian sigma function with its argument shifted by the Abelian image of the
reduced divisor of a single point on the curve. The genus two recurrences are
related to a B\"{a}cklund transformation (BT) for an integrable Hamiltonian
system, namely the discrete case (ii) H\'{e}non-Heiles system.
