An infinite family of `Kac-Moody Fibonacci' integer sequences

[Lisa Carbone]

 In the paper https://arxiv.org/pdf/2601.00958, we summarize known results on how to generate infinite families of integer sequences from the coefficients of real roots on 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. 

For example,  define a sequence X(k) = x_0, x_1, x_2, ... by:

 x_0 = 0,  x_1 = 1 , x_n = k x_{n-1} - x_{n-2} for  n >= 2.

Then X(a) = 0, 1, a, a^2-1, a^3-2a, a^4-3a^2+1,... and X(3)= 0, 1, 3, 8, 21, ... is a bisection of the Fibonacci sequence.

There are other families of sequences

X_1(a,b)=1, ab-1, a^2b^2-3ab+1,...

X_2(a,b)=0, a, a^2b-2a, a^3b^2-4a^2b+3a,...

which also satisfy recursions. Many of these sequences are in OEIS. For example X(1,5)= 1, 4, 11, 29, 76, ... is a bisection of the Lucas sequence. The OEIS description is also tabulated in our paper. We present many new sequences not in OEIS as well.