{"id":2643,"date":"2025-07-26T17:27:17","date_gmt":"2025-07-26T17:27:17","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=2643"},"modified":"2025-07-26T21:27:02","modified_gmt":"2025-07-26T21:27:02","slug":"squares-of-fibonacci-like-numbers","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/squares-of-fibonacci-like-numbers\/","title":{"rendered":"Squares of Fibonacci-like Numbers"},"content":{"rendered":"\n
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Moroccan Journal of Algebra and Geometry with Applications<\/strong><\/p>\n\n\n\n

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Squares of Fibonacci-like Numbers<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

Kunle Adegoke<\/strong>\\,^1<\/span><\/i> and Tokunbo Omiyinka\\,^2<\/span>
\\,^{1}<\/span>Department of Physics and Engineering Physics, Obafemi Awolowo University, Ile-Ife, Nigeria
\\,^{2}<\/span>Department of Mathematical and Physical Sciences, Concordia University of Edmonton
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Faculty of Education, University of Alberta, Edmonton, Alberta<\/p>\n\n\n\n

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Pages 62-73 | Received 10 June 2023, Accepted 27 November 2024, Published 10 July 2025 <\/span><\/p>\n\n\n\n

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Abstract<\/span><\/strong><\/p>\n\n\n\n

Given arbitrary integers G_0<\/span> and [\/katex]G_1[\/katex], not both zero, Fibonacci-like numbers, G_j<\/span>, are defined for all non-megative integers j<\/span> by the recurrence relation G_j=G_{j-1}+G_{j-2}\\; (j \\ge 2)<\/span>. 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}<\/span> and \\displaystyle\\displaystyle\\sum_{j=0}^n{G_{j+k}G_{j-k}}<\/span>. We extend these results by providing evaluations for \\displaystyle\\displaystyle\\sum_{j=0}^n{x^jG_{j+k}^2}<\/span> and \\displaystyle\\displaystyle\\sum_{j=0}^n{x^jG_{j+k}G_{j+s}}<\/span> for integers n, \\ s<\/span> and k<\/span> and a real or complex variable x<\/span>. 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.<\/p>\n\n\n\n

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Keywords<\/strong>:<\/span>  Fibonacci number, Lucas number, Fibonacci-like number,vgenerating function.<\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> Primary 11B39; Secondary 11B37.<\/p>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles Squares of Fibonacci-like Numbers Kunle Adegoke and Tokunbo Omiyinka Department of Physics and Engineering Physics, Obafemi Awolowo University, Ile-Ife, Nigeria 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 Read More<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-page-builder.php","meta":{"footnotes":""},"class_list":["post-2643","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2643","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/comments?post=2643"}],"version-history":[{"count":4,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2643\/revisions"}],"predecessor-version":[{"id":2701,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2643\/revisions\/2701"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=2643"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}