Kac--Moody Fibonacci sequences

We summarize known results on how to generate an infinite family of integer sequences from the root lattices of rank 2 Kac–Moody algebras. We compute and tabulate the first twenty entries of a number of these sequences. This provides an overarching framework for a large class of Fibonacci-type integer sequences, evaluations of Chebyshev S and U-polynomials and others.