On k-Fibonacci numbers expressible as product of two Balancing or Lucas-Balancing numbers

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 8n(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.