01/05/2026
This research explored mathematical extensions of the patterns commonly seen in nature, such as the Fibonacci and Lucas sequences. By generalizing these rules into a "Horadam sequence," the researchers introduced a bi-periodic parameter that alternates between two values based on the term's position. This innovative approach led to the discovery of the bi-periodic Fibonacci-Horadam and Lucas-Horadam sequences. Using matrix techniques, the authors uncovered new properties of these generalized mathematical structures.
Authors: Joy P. Ascano (Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio), Edna N. Gueco (Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio), and Julius Fergy T. Rabago (Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University)
Published in the Communications of the Korean Mathematical Society
Read more: https://ovpaa.up.edu.ph/research/research-explores-extensions-to-the-fibonacci-and-lucas-sequences/
RESEARCH | If you observe nature, objects such as pinecones, pineapples, and sunflowers have a spiral-like structure. Interestingly, if you count these spirals, they seem to give you random numbers. For example, a pinecone may have 8 and 13 spirals, and a sunflower even larger numbers like 34 and 55. These numbers may appear random at first, but they occur repeatedly in nature. They follow a mathematical sequence that is 1,1,2,3,5,8,13,21,34,55, and so on. This sequence is called the Fibonacci sequence. What’s fascinating about the Fibonacci sequence is that it follows a simple rule, which can be expressed as an equation: F_n=F_(n-1)+F_(n-2).
In simple terms, any number in the sequence is the sum of the two preceding numbers: 1+2 is 3, 2+3 is 5, 3+5 is 8, and so on. However, if you observe more closely (in nature), you may also find spiral patterns that don’t follow the Fibonacci numbers. Some pinecones may have 7 and 11 spirals. But on closer inspection, these numbers also follow the same equation but start with different numbers. This new sequence is called the Lucas sequence, which goes 2, 1, 3, 4, 7, 11, 18, 29, and so on. Like the Fibonacci sequence, each Lucas number is the sum of the two numbers before it.
The ubiquity of these mathematical sequences in nature has long intrigued mathematicians about the properties of these numbers. In our research, we focus on discovering additional properties of these sequences. To achieve this, we introduced a more general equation by inserting parameters into both terms. This general form is known as the Horadam sequence. In our study, we let the first parameter be bi-periodic, meaning it alternates between two values depending on whether the term is in an odd or even position. This led to the bi-periodic Fibonacci-Horadam and bi-periodic Lucas-Horadam sequences. We apply the concept of matrices (arrays of numbers) and demonstrate how matrix techniques can uncover new properties.
Authors: Joy P. Ascano (Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio), Edna N. Gueco (Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio), and Julius Fergy T. Rabago (Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University)
Published in the Communications of the Korean Mathematical Society
Read more: https://ovpaa.up.edu.ph/research/research-explores-extensions-to-the-fibonacci-and-lucas-sequences/
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