# What is the Fibonacci sequence? ~ A mysterious sequence also exists in nature ~

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233… Do you know what the rules are for these numbers? This sequence is called the Fibonacci sequence. Let’s examine this sequence, which has a mysterious property.

# What are the rules for the Fibonacci sequence?

The Fibonacci sequence has the regularity that “**the sum of the previous two terms becomes the next term**”. For example, look at the first three terms that appear, 1 + 1 = 2. Similarly, 2 to 3 terms are 1 + 2 = 3, 3 to 3 terms are 2 + 3 = 5 … and so on, followed by a regular sequence.

# What is Fibonacci (Pisa’s Leonardo) from which the name was derived?

Fibonacci, the origin of the Fibonacci sequence, is a medieval Italian mathematician, also known as Fibonacci of Pisa. When Fibonacci was young, he traveled to Byzantine (Turkey), Sicily, Egypt, Syria and other eastern Mediterranean coastal regions for his father’s business, where he studied advanced Arabic mathematics and returned to pizza in 1200 AD and published a compilation of what he learned. However, European mathematics at that time was still behind, and it took more than 200 years till Fibonacci’s writings were accepted. The Fibonacci sequence was also written in the book, and the original story is said to be in the old Indian Sanskrit lyrics. In Addition, this sequence came to the forefront more than 600 years after Fibonacci’s writing. The mathematician named Edouard Lucas named this sequence the Fibonacci sequence. Fibonacci’s writings have the following story:

# Fibonacci tree. How many rabbits will be in a year?

A farmer got a pair of rabbits. Rabbits grow up in one month and give birth to children just before the second month. One pair of rabbits gives birth to one pair of baby rabbits every month. How many rabbits will be in a year’s time? One pair means one male and one female, for a total of two animals.

This is called **the Fibonacci tree**. The top apex is called the root of this tree. Oddly enough, in mathematics, we write trees “upside down” like this. In other words, the branches come out downward from each vertex. This tree has one or two branches coming out of the apex. The apex without the bottom branch is called a leaf. The top vertex (ie root) is level 0, the next vertex is level 1, and the levels are 0, 1, 2,… from top to bottom.

Let uₙ represent the number of vertices at level n. Then

## u₀ , u₁ , u₂ , …

is the Fibonacci sequence mentioned above. Interestingly, the Fibonacci sequence is also in parentheses.

The uₙ of uₙ (uₙ₋₁) written at level n represents the number of pairs after n months, and uₙ₋₁ represents the number of pairs of parent rabbits in it. Therefore, the number of pairs of baby rabbits is uₙ — uₙ₋₁. Of the uₙ pairs, only the parent pair uₙ₋₁ gives birth to a child, and the number of pairs increases by uₙ₋₁, so the number of pairs in the (n + 1) month uₙ₊₁ is “uₙ”. ₊₁ = uₙ + uₙ₋₁ “. In other words, it satisfies the rule that “the sum of the previous two terms becomes the next term”.

The sequence defined by Fibonacci was 1, 2, 3, 5, 8, …, but first added 1. Nowadays, the Fibonacci sequence may be a sequence of 1 plus 0.

The Fibonacci sequence is a sequence represented by the following.

**u₁ = 1, u₂ = 1, uₙ₊₁ = uₙ + uₙ₋₁ (n = 2, 3, 4,…)**

The Fibonacci sequence often appears in unexpected places. Not only artificial ones, but Fibonacci sequences are ubiquitous in nature. It seems that the Fibonacci sequence appears in nature **because of a kind of “self-similarity”**. In other words, as you grow, you can save the information you need by constructing something similar to what you’ve made so far.

Happy Science life !