Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences

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.