On Fibonacci numbers as sum of powers of two consecutive Tribonacci numbers.

For k ≥ 2 , the sequence (F n (k)) n ≥ - (k - 2) of k-generalized Fibonacci numbers is defined by the initial values 0 , … , 0 , 1 = F 1 (k) and such that each term afterwards is the sum of the k preceding ones. There are many recent results about the Diophantine equation (F n (k)) s + (F n + 1 (k)) s = F m (ℓ) , most of them dealing with the case k = ℓ . In 2018, Bednařík et al. solved the equation for k ≤ ℓ , but with s = 2 . The aim of this paper is to continue this line of investigation by solving this equation for all s ≥ 2 , but with (k , ℓ) = (3 , 2) . [ABSTRACT FROM AUTHOR]