Common values of generalized Fibonacci and Pell sequences

Abstract

A generalization of the well–known Fibonacci sequence is the ℓ-Fibonacci sequence (Fm(ℓ))m whose first ℓ terms are 0,…,0,1 and each term afterwards is the sum of the preceding ℓ terms. The k-Pell sequence (Pn(k))n, which is a generalization of the classical Pell sequence, can be defined similarly. In this paper, we find all coincidences between these two families of sequences. That is, we find all the solutions of the Diophantine equation Pn(k)=Fm(ℓ) in positive integers n,k,m,ℓ with k,ℓ≥2. This paper continues and extends prior results which dealt with the above problem for some particular cases of k and ℓ. In particular, it extends the previous work [2] that found all Fibonacci numbers in the Pell sequence.