Counting the sum of cubes for Lucas and Gibonacci numbers

Abstract

Lucas and Gibonacci numbers are  two  sequences of numbers derived from a welknown numbers, Fibonacci numbers. The difference between Lucas and Fibonacci numbers only lies on the first and second elements.  The first element in Lucas numbers  is 2 and the second is 1, and n^th element,  n ≥ 3 determined by similar pattern as in the Fibonacci numbers, i.e : L_n = L_n-1  + L_n-2.  Gibonacci numbers  G₀ , G₁ ,G₂ , ...; G_n = G_n-1 + G_n-2 are generalized of Fibonacci numbers, and those numbers are nonnegative integers. If G₀ = 1 and G₁ = 1, then the numbers are the wellknown Fibonacci numbers, and if G₀ = 2 and G₁ = 1, the numbers are Lucas numbers. Thus, the difference of those three sequences of numbers only lies on the first and second of the elements in the sequences.  For Fibonacci numbers there are quite a lot identities already explored, including the sum of cubes, but there have no discussions yet about the sum of cubes for Lucas and Gibonacci numbers. In this study the sum of  cubes of Lucas and Gibonacci numbers will be discussed and showed that the sum of cubes for Lucas numbers  is   and for Gibonacci numbers is