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In this paper, we develop a new formula for hyper-Fibonacci numbers F [k] n , wherein the coefficients (related to Stirling numbers of the first kind) of the polynomial ingredient pk(n) are determined. As an application we investigate the number of occurrences of positive integers among F [k] n and determine all the solutions in nonnegative integers x and y to the Diophantine equation F [k] x = F [l] y , where 0 ≤ k < l ≤ 70. Moreover, we prove that if l is fixed, then F [k] x = F [l] y has finitely many effectively computable solutions in the nonnegative integers x , y , and k ≤ l .

A new formula for hyper-Fibonacci numbers, and the number of occurrences