  We investigate a general method that allows one to construct new integer
sequences extending existing ones. We apply this method to the classic Somos-4
and Somos-5, and the Gale-Robinson sequences, as well as to more general class
of sequences introduced by Fordy and Marsh, and produce a great number of new
sequences. The method is based on the notion of "weighted quiver", a quiver
with a $\mathbb Z$-valued function on the set of vertices that obeys very
special rules of mutation.
