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

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 ∈ [ 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) ∈ Z ≥ r + 2 × Z ≥ 1 . [ABSTRACT FROM AUTHOR]