Is the Fibonacci sequence the only increasing strong divisibility sequence such that u_0=0 and u_1=1?

Question

The Fibonacci sequence has several interesting properties, among which the following ones :

• $ F_{0}=0 $ and $ F_{1}=1 $

• $ (F_n) $ is a strong divisibility sequence

• $ (F_n) $ is an increasing sequence.

Are there any other integer sequences fulfilling all those properties ?

Answer 1

Any recurrence relation of the form

$$F_0=0, F_1=1, \quad F_n= F_{n-1}+k\cdot F_{n-2}$$

where $k$ is a positive integer. Fibonacci is the case with $k=1$.

Also, any sequence given by $\frac{p^k-1}{p-1}$ where $p,k$ are coprime.

Answer 2

This article might be of interest to you:[https://www.hindawi.com/journals/isrn/2014/750325/ ]. It specifically mentions a generalization of the Fibonacci sequence with $L_{n+1}=pL_{n}+qL_{n-1}$ and $gcd(p,q)=1$ called the Generalized Lucas Sequence. They also prove that it is a strong devisibility sequence and it is obviously increasing.