Double-Recurrence Fibonacci Numbers and Generalizations

Abstract

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by the recurrence $F_{n+2}=F_{n+1}+F_n$, for $n\geq 0$, where $F_0=0$ and $F_1=1$. There are several generalizations of this sequence and also several interesting identities. In this paper, we investigate a homogeneous recurrence relation that, in a way, extends the linear recurrence of the Fibonacci sequence for two variables, called {\it double-recurrence Fibonacci numbers}, given by ${F(m,n) = F(m-1, n-1)+F (m-2, n-2)}$, for $n,m\geq 2$, where $F (m, 0) = F_m$, $F (m, 1) = F_{m+1}$, $F (0, n) = F_n$ and $F (1, n) = F_{n+1}$. We exhibit a formula to calculate the values of this double recurrence, only in terms of Fibonacci numbers, such as certain identities for their sums are outlined. Finally, a general case is studied.