Squares of Fibonacci-like Numbers
Kunle Adegoke\,^1 and Tokunbo Omiyinka\,^2
\,^{1}Department of Physics and Engineering Physics, Obafemi Awolowo University, Ile-Ife, Nigeria
\,^{2}Department of Mathematical and Physical Sciences, Concordia University of Edmonton
and
Faculty of Education, University of Alberta, Edmonton, Alberta
Pages 62-73 | Received 10 June 2023, Accepted 27 November 2024, Published 10 July 2025
Abstract
Given arbitrary integers G_0 and [/katex]G_1[/katex], not both zero, Fibonacci-like numbers, G_j, are defined for all non-megative integers j by the recurrence relation G_j=G_{j-1}+G_{j-2}\; (j \ge 2). In this paper, we derive a general identity involving the squares of Fibonacci-like numbers. Closed formulas exist for \displaystyle\displaystyle\sum_{j=0}^n{x^jG_{j}^2} and \displaystyle\displaystyle\sum_{j=0}^n{G_{j+k}G_{j-k}}. We extend these results by providing evaluations for \displaystyle\displaystyle\sum_{j=0}^n{x^jG_{j+k}^2} and \displaystyle\displaystyle\sum_{j=0}^n{x^jG_{j+k}G_{j+s}} for integers n, \ s and k and a real or complex variable x. Various other properties are developed, including double binomial summation identites. As a bonus unexpected result we derive a generalization of the alternating sum of the products of three consecutive Fibonacci-like numbers.
Keywords: Fibonacci number, Lucas number, Fibonacci-like number,vgenerating function.
MSC numbers: Primary 11B39; Secondary 11B37.
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