On k-Fibonacci balancing and k-Fibonacci Lucas-balancing numbers - Rihane, S.E. (Carpathian Mathematical Publications, 2021)
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. For an integer k≥ 2, let (F_n^(k))_n be the k-generalized Fibonacci sequence which starts with 0,…,0,1,1 (k terms) and each term afterwards is the sum of the k preceding terms. The purpose of this paper is to show that 1, 6930 are the only balancing numbers and 1, 3 are the only Lucas-balancing numbers which are a term of k-generalized Fibonacci sequence. This generalizes the result from [Fibonacci Quart. 2004, 42 (4), 330-340].