On the Natural Density of Sets Related to Generalized Fibonacci Numbers of Order r

Abstract

For r >= 2 and a >= 1 integers, let (t(n)((r,a)))(n >= 1) be the sequence of the (r,a)-generalized Fibonacci numbers which is defined by the recurrence t(n)((r,a))=t(n-1)((r,a))+ . . .+t(n-r)((r,a)) for n>r, with initial values t(i)((r,a))=1, for all i is an element of[1,r-1] and t(r)((r,a))=a. In this paper, we shall prove (in particular) that, for any given r >= 2, there exists a positive proportion of positive integers which can not be written as t(n)((r,a)) for any (n,a)is an element of Z(>= r+2)xZ(>1).