Research explores extensions to the Fibonacci and Lucas sequences

30 Apr 2026

If you observe nature, objects, such as pinecones, pineapples, and sunflowers, have a spiral-like structure. Often, these spiral patterns can be seen in two directions (sometimes more). Interestingly, if you count these spirals, they seem to give you random numbers. For example, a pinecone may have 8 and 13 spirals, a pineapple 13 and 21, and a sunflower even larger numbers like 34 and 55. These numbers may appear random at first, but they occur repeatedly in nature. They interestingly 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, while others might have 11 and 18. 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. One remarkable property is that the ratio of consecutive terms in the Fibonacci sequence approaches an irrational number, (1+sqrt{5})/2, known as the golden ratio, a value that also appears frequently in natural patterns. Why and how plants exhibit such growth patterns is a question rooted in biology, but that is beyond the scope of this discussion. Our focus lies in the fascinating mathematical properties these sequences display.

For example, the summation of the first few terms in the Fibonacci or Lucas sequences is one less than the second term after the last one added. That is, 1+1+2+3+5+8+13+21 is 55-1 and 2+1+3+4+7+11 is 29-1.

Over the years, mathematicians have explored what happens when parameters are introduced into these equations. This led to variations like the Pell and Jacobsthal sequences. These were studied by mathematicians such as John Pell and Ernst Jacobsthal, who modified the recurrence formulas as follows:
P_n=2P_(n-1)+P_(n-2) and J_n=J_(n-1)+2J_(n-2).

Here, instead of simply adding the two previous terms, one of them is doubled before being added. The Pell sequence begins as: 1,2,5,12,29,70, and so on, while the Jacobsthal sequence is given by 1,1,3,5,11,21,43, and so on.

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 specifically 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.

Recent studies have explored various methods to analyze such sequences. In this paper, we apply the concept of matrices (arrays of numbers) and demonstrate how matrix techniques can be used to uncover new properties.

One of our key contributions is a generalization of the golden ratio and an analysis of how the ratios of consecutive terms in these sequences converge to it. Additionally, we introduced a new matrix tailored to the bi-periodic Lucas-Horadam sequence, which allowed us to derive further identities, including generalized summation formulas.

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)

Read the full paper: http://doi.org/10.4134/CKMS.c240121

Image by Carolina Prado from Pexels

Research explores extensions to the Fibonacci and Lucas sequences

If you observe nature, objects, such as pinecones, pineapples, and sunflowers, have a spiral-like structure. Often, these spiral patterns can be seen in two directions (sometimes more). Interestingly, if you count these spirals, they seem to give you random numbers. For example, a pinecone may have 8 and 13 spirals, a pineapple 13 and 21, and a sunflower even larger numbers like 34 and 55. These numbers may appear random at first, but they occur repeatedly in nature. They interestingly 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, while others might have 11 and 18. 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. One remarkable property is that the ratio of consecutive terms in the Fibonacci sequence approaches an irrational number, (1+sqrt{5})/2, known as the golden ratio, a value that also appears frequently in natural patterns. Why and how plants exhibit such growth patterns is a question rooted in biology, but that is beyond the scope of this discussion. Our focus lies in the fascinating mathematical properties these sequences display.

For example, the summation of the first few terms in the Fibonacci or Lucas sequences is one less than the second term after the last one added. That is, 1+1+2+3+5+8+13+21 is 55-1 and 2+1+3+4+7+11 is 29-1.

Over the years, mathematicians have explored what happens when parameters are introduced into these equations. This led to variations like the Pell and Jacobsthal sequences. These were studied by mathematicians such as John Pell and Ernst Jacobsthal, who modified the recurrence formulas as follows:
P_n=2P_(n-1)+P_(n-2) and J_n=J_(n-1)+2J_(n-2).

Here, instead of simply adding the two previous terms, one of them is doubled before being added. The Pell sequence begins as: 1,2,5,12,29,70, and so on, while the Jacobsthal sequence is given by 1,1,3,5,11,21,43, and so on.

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 specifically 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.

Recent studies have explored various methods to analyze such sequences. In this paper, we apply the concept of matrices (arrays of numbers) and demonstrate how matrix techniques can be used to uncover new properties.

One of our key contributions is a generalization of the golden ratio and an analysis of how the ratios of consecutive terms in these sequences converge to it. Additionally, we introduced a new matrix tailored to the bi-periodic Lucas-Horadam sequence, which allowed us to derive further identities, including generalized summation formulas.

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)

Read the full paper: http://doi.org/10.4134/CKMS.c240121

Image by Carolina Prado from Pexels