Research

As the national university, we champion and support innovative research that addresses the country’s most pressing challenges.

30 Apr 2026

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...

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29 Apr 2026

Correctional officers address incarcerated persons with dignifying language but often revert to stigmatizing labels in private staff discussions

Language profoundly shapes perceptions, particularly within the confines of prisons. This study, which analyzed data from the Scandinavian Prison Project...

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28 Apr 2026

Study looks at how scientists have been able to extract nanocellulose from banana peel, turning waste into something useful

Banana peels (BP) are often thrown away as waste, but they actually contain valuable natural fibers called nanocellulose (NC). These...

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27 Apr 2026

Yerba buena has a wide range of potential health benefits, including antibacterial, pain relieving, and anticancer properties

Yerba buena (Mentha × villosa Huds., family Lamiaceae) is one of the ten herbal medicines endorsed by the Department of...

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24 Apr 2026

Social media, especially when guided by healthcare professionals, can be a powerful tool for patient education and emotional support

Thyroid conditions affect millions of people worldwide and can greatly affect daily life. Since managing these conditions usually takes a...

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22 Apr 2026

Men are likely to practice yoga when they are supported and encouraged by their family, friends, and partners

Despite the growing popularity of yoga, men remain underrepresented in yoga classes and activities. This research aimed to investigate the...

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21 Apr 2026

Areas of the Abra River watershed, particularly near human settlements, show signs of declining ecological health

The Abra River Basin and its associated watershed is one of the 18 major river systems in the country, flowing...

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17 Apr 2026

A community health club has seen improvements in the number of members with hypertension but not in those with diabetes

From 2013 to 2021, the UP Manila Community Health and Development Program partnered with the province of Cavite and its...

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16 Apr 2026

Tensions and disagreements within indigenous communities over representation and creative innovation influence indigenous traditional practices

Street protests led by Igorot left-wing activists feature sharp rhythms of gangsa (flat gongs), political speeches over loudspeakers, and chants...

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15 Apr 2026

Clear communication, strong mentorship, and approachable faculty help reduce student stress and enhance engagement

College students in health-related courses, like medicine, nursing, and occupational and physical therapy, often deal with stress, burnout, and mental...

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14 Apr 2026

Study examines the latest Small Modular Reactor technologies and which designs may be considered for use in the Philippines

As the Philippines continues to grow, our need for electricity is also increasing. At the same time, our natural gas...

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10 Apr 2026

Using machine learning, study finds that key factors influencing poverty include country, whether people live in urban or rural areas, and their level of education

Relying only on income to determine poverty is not enough to fully understand someone’s well-being. Other factors, such as where...

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Research

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

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