On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers

Abstract

Let (t(n)((r)))(n >= 0) be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence t(n)((r)) = t(n-1)((r)) +... + t(n-r)((r)) for n >= r, with initial values t(0)((r)) = 0 and t(i)((r)) = 1, for all 1 <= i <= r. In 2002, Grossman and Luca searched for terms of the sequence (t(n)((2)))(n), which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any l >= 1, there exists an effectively computable constant C = C(l) > 0 (only depending on l), such that, if (m, n, r) is a solution of t(m)((r)) = n! + (n + 1)! +... + (n + l)!, with r even, then max{m, n, r} < C. As an application, we solve the previous equation for all 1 <= l <= 5.