On k-Fibonacci numbers expressible as product of two Balancing or Lucas-Balancing numbers: On k-Fibonacci numbers expressible...: S. E. Rihane.

The Balancing number n and the balancer r are solution of the Diophantine equation 1 + 2 + ⋯ + (n - 1) = (n + 1) + (n + 2) + ⋯ + (n + r) . It is well known that if n is balancing number, then 8 n 2 + 1 is a perfect square and its positive square root is called a Lucas-Balancing number. Let k ≥ 2 . A generalization of the well-known Fibonacci sequence is the k-Fibonacci sequences. For these sequence the first k terms are 0 , ... , 0 , 1 and each term afterwards is the sum of the preceding k terms. In this manuscript, our main objective is to find all k-Fibonacci numbers which are the product of two Balancing or Lucas-Balancing numbers. [ABSTRACT FROM AUTHOR]