Version 1
: Received: 22 April 2020 / Approved: 24 April 2020 / Online: 24 April 2020 (10:35:07 CEST)

How to cite:
Soykan, Y. A Study On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of $\sum_{k=0}^{n}x^{k}W_{k}^{3}$ and $\sum_{k=1}^{n}x^{k}W_{-k}^{3} $. Preprints2020, 2020040437 (doi: 10.20944/preprints202004.0437.v1).
Soykan, Y. A Study On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of $\sum_{k=0}^{n}x^{k}W_{k}^{3}$ and $\sum_{k=1}^{n}x^{k}W_{-k}^{3} $. Preprints 2020, 2020040437 (doi: 10.20944/preprints202004.0437.v1).

Cite as:

Soykan, Y. A Study On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of $\sum_{k=0}^{n}x^{k}W_{k}^{3}$ and $\sum_{k=1}^{n}x^{k}W_{-k}^{3} $. Preprints2020, 2020040437 (doi: 10.20944/preprints202004.0437.v1).
Soykan, Y. A Study On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of $\sum_{k=0}^{n}x^{k}W_{k}^{3}$ and $\sum_{k=1}^{n}x^{k}W_{-k}^{3} $. Preprints 2020, 2020040437 (doi: 10.20944/preprints202004.0437.v1).

Abstract

In this paper, closed forms of the sum formulas $\sum_{k=0}^{n}x^{k}W_{k}^{3}$ and $\sum_{k=1}^{n}x^{k}W_{-k}^{3}$ for the cubes of generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers.

Subject Areas

Fibonacci numbers; Lucas numbers; Pell numbers; Jacobsthal numbers; sum formulas

Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.