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

## TL;DR

This paper introduces a novel method using weighted quivers to generate new integer sequences extending classic Somos and Gale-Robinson sequences, broadening the scope of sequence construction.

## Contribution

It presents a new approach based on weighted quivers with special mutation rules to systematically create extended integer sequences.

## Key findings

- Generated numerous new sequences from classic ones.
- Established a framework connecting quivers with sequence extension.
- Demonstrated the method's applicability to various sequence classes.

## Abstract

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.