Double-Recurrence Fibonacci Numbers and Generalizations

Abstract

Let $(F_n{)}_{n\ge 0}$ be the Fibonacci sequence given by the recurrence $F_{n+2}=F_{n+1}+F_n$, for $n\ge 0$, 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 $n,m\ge 2$, 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.

10 pages, 5 figures and 1 table