Fibonacci sequence facts for kids

Kids Encyclopedia Facts

The Fibonacci sequence is a fascinating list of numbers where each number is the sum of the two numbers right before it. These special numbers are called Fibonacci numbers. We often write them as Fn, where n tells us its place in the sequence. The sequence usually starts with F0 = 0, F1 = 1, and F2 = 1.

Starting from F0, the sequence looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

This picture shows squares whose side lengths are Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, and 21.

Fibonacci numbers were first described in Indian mathematics a very long time ago, around 200 BC. An Indian scholar named Pingala used them to study patterns in Sanskrit poetry. Later, an Italian mathematician named Fibonacci (whose real name was Leonardo of Pisa) introduced these numbers to Europe in his book Liber Abaci in 1202.

You might be surprised how often Fibonacci numbers pop up in math! There's even a whole magazine, the Fibonacci Quarterly, just for studying them. They are used in computer programs, like special search methods and ways to organize data. They also appear in nature, such as how branches grow on trees, how leaves are arranged on a stem, the patterns on a pineapple, the petals of an artichoke, and the scales of a pine cone.

Fibonacci numbers are also closely connected to something called the golden ratio. As you go further into the sequence, the ratio of one Fibonacci number to the one before it gets closer and closer to this golden ratio.

What are Fibonacci Numbers?

This is the Fibonacci spiral. It's a close copy of the golden spiral, made by drawing curved lines inside the Fibonacci squares.

We can define Fibonacci numbers with a simple rule:

The first number, F0, is 0.
The second number, F1, is 1.
For any number after that, Fn is found by adding the two numbers before it: Fn = Fn-1 + Fn-2.

So, if we want to find F3, we add F2 and F1 (1 + 1 = 2). To find F4, we add F3 and F2 (2 + 1 = 3), and so on.

Here are the first 21 Fibonacci numbers:

F0	F1	F2	F3	F4	F5	F6	F7	F8	F9	F10	F11	F12	F13	F14	F15	F16	F17	F18	F19	F20
0	1	1	2	3	5	8	13	21	34	55	89	144	233	377	610	987	1597	2584	4181	6765

We can even extend the Fibonacci sequence to include negative numbers. The rule still works! For example, F-1 would be F1 - F0 = 1 - 0 = 1.

A Look at History

From Ancient India

This image shows 13 ways to arrange long and short syllables in a six-beat rhythm. This is an example of Fibonacci numbers in ancient Indian poetry.

The Fibonacci sequence first appeared in Indian mathematics when people were studying Sanskrit poetry. Poets were interested in counting different patterns of long and short syllables. A long syllable took 2 units of time, and a short syllable took 1 unit. If you counted the number of ways to make a rhythm of a certain total length, you would find Fibonacci numbers!

One of the earliest mentions was by Pingala around 450-200 BC. Later, scholars like Virahanka (around 700 AD) and Hemachandra (around 1150 AD) also wrote about this sequence, describing how to find the next number by adding the previous two.

Fibonacci's Rabbits in Europe

A page from Fibonacci's book, Liber Abaci. You can see 13 Fibonacci numbers listed, showing the growth of rabbit pairs.

The sequence became known in Europe thanks to Fibonacci and his book Liber Abaci in 1202. He used it to solve a famous problem about how fast a population of rabbits would grow.

Imagine you start with one new pair of rabbits.

After one month, they mate. You still have 1 pair.
After two months, they have a new pair. Now you have 2 pairs.
After three months, the first pair has another new pair. The second pair is now old enough to mate. You have 3 pairs.
After four months, the first pair has another new pair, and the second pair has their first pair. Now you have 5 pairs!

This diagram shows how the number of rabbit pairs grows each month, following the Fibonacci sequence.

If you keep tracking, the number of rabbit pairs each month follows the Fibonacci sequence: 1, 1, 2, 3, 5, 8, and so on. The name "Fibonacci sequence" was given much later by a mathematician named Édouard Lucas in the 19th century.

The Golden Ratio Connection

The Fibonacci numbers have a special relationship with the golden ratio, which is a number approximately equal to 1.618.

Approaching the Golden Ratio

This animation shows how squares with Fibonacci side lengths can tile a plane. The graph shows how the ratio of consecutive Fibonacci numbers gets closer to the golden ratio.

A famous astronomer named Johannes Kepler noticed something amazing: if you divide a Fibonacci number by the one before it, the answer gets closer and closer to the golden ratio. For example:

8 ÷ 5 = 1.6
13 ÷ 8 = 1.625
21 ÷ 13 = 1.615...
34 ÷ 21 = 1.619...

As you go further in the sequence, these ratios get incredibly close to the golden ratio (about 1.61803...).

Cool Math Tricks with Fibonacci Numbers

Counting Paths and Combinations

Fibonacci numbers can help us count things in different ways! Imagine you want to find the number of ways to write a number as a sum of only 1s and 2s.

For the number 1, there's 1 way: (1)
For the number 2, there are 2 ways: (1,1), (2)
For the number 3, there are 3 ways: (1,1,1), (1,2), (2,1)
For the number 4, there are 5 ways: (1,1,1,1), (1,1,2), (1,2,1), (2,1,1), (2,2)

Notice a pattern? The number of ways to sum to n is Fn+1!

This image shows how Fibonacci numbers can help count the ways to climb stairs, taking one or two steps at a time.

This is like climbing stairs! If you can take either 1 or 2 steps at a time, the number of ways to climb a staircase of n steps is Fn+1. For 5 steps, there are F6 = 8 ways:

1+1+1+1+1
2+1+1+1
1+2+1+1
1+1+2+1
1+1+1+2
2+2+1
2+1+2
1+2+2

Fibonacci in Pascal's Triangle

The Fibonacci numbers appear as the sums of the diagonal rows in Pascal's triangle.

You can also find Fibonacci numbers hidden in Pascal's triangle! If you add up the numbers along certain diagonals in Pascal's triangle, you'll get Fibonacci numbers. It's like a secret code!

Sums of Squares

There's a neat identity: if you add up the squares of the first few Fibonacci numbers, you get the product of the last Fibonacci number and the next one. For example: F1² + F2² + F3² + F4² = 1² + 1² + 2² + 3² = 1 + 1 + 4 + 9 = 15. And F4 × F5 = 3 × 5 = 15. It works!

A visual proof that the sum of squares of Fibonacci numbers equals the product of two consecutive Fibonacci numbers.

Fibonacci and Prime Numbers

Divisibility Rules

Fibonacci numbers have cool divisibility rules. Every third Fibonacci number is even (like F3 = 2, F6 = 8, F9 = 34). More generally, if you pick any Fibonacci number Fk, then every Fn where n is a multiple of k will also be a multiple of Fk.

Also, any two consecutive Fibonacci numbers (like Fn and Fn+1) don't share any common factors other than 1. They are coprime.

Fibonacci Primes

A Fibonacci prime is a Fibonacci number that is also a prime number. The first few are: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... Scientists are still trying to figure out if there are infinitely many Fibonacci primes.

Did you know that the only Fibonacci numbers that are perfect squares are 1 (F1 and F2) and 144 (F12)? Also, the only Fibonacci numbers that are triangular numbers are 1, 3, 21, and 55.

Repeating Patterns

If you look at the Fibonacci sequence and only care about the remainder when you divide by a certain number n (this is called "modulo n"), the sequence will always repeat in a pattern. The length of this repeating pattern is called a Pisano period. For example, if you look at the Fibonacci numbers modulo 2, the sequence is 0, 1, 1, 0, 1, 1, ... and the pattern (0, 1, 1) repeats every 3 numbers.

Other Fibonacci-like Sequences

The Fibonacci sequence is just one example of a sequence where each number depends on the ones before it. There are many others!

Lucas numbers are similar but start with 1, 3, 4, 7, 11...
Pell numbers use the rule Pn = 2Pn-1 + Pn-2.
Tribonacci numbers add the three previous numbers instead of two.

Where We See Fibonacci Numbers

In Mathematics

Counting with 1s and 2s: As we saw, Fibonacci numbers help count how many ways you can make a number by adding 1s and 2s.
Binary Strings: They also count special kinds of binary codes (strings of 0s and 1s). For example, the number of binary strings of length n that don't have two 1s next to each other is Fn+2.
Pythagorean Triples: Starting with 5, every second Fibonacci number can be the longest side (hypotenuse) of a right triangle with whole number sides! For example, (3,4,5) and (5,12,13).

In Computer Science

This is a Fibonacci tree of height 6. These trees are important in computer science for organizing data.

Efficient Algorithms: Fibonacci numbers are used in understanding how fast some computer programs work, like Euclid's algorithm for finding the greatest common divisor.
Sorting Data: They are used in a clever way to sort lists of information, called a merge sort.
Data Structures: Special ways to organize data in computers, like a Fibonacci heap, use these numbers.
Planning Poker: Some software development teams use a modified Fibonacci sequence to estimate how much work a task will take.

In Nature

A yellow chamomile flower head showing spirals of 21 (blue) and 13 (cyan) petals. These numbers are consecutive Fibonacci numbers.

Fibonacci numbers appear all over the natural world!

Plant Growth: You can see them in how branches grow on trees, how leaves are arranged on a stem (called Phyllotaxis), and the patterns of seeds in a sunflower or scales on a pine cone. These patterns often involve spirals where the number of spirals in one direction and the other are consecutive Fibonacci numbers.

This model shows the pattern of florets (small flowers) in a sunflower head.

Bee Family Trees: The family tree of male honeybees follows the Fibonacci sequence. A male bee has one parent (a queen bee). That queen bee has two parents (a queen and a drone). So, a male bee has 1 parent, 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on!

The number of ancestors on the human X chromosome line also follows the Fibonacci sequence.

Human Ancestry: The number of ancestors who contributed to a male's X chromosome also follows the Fibonacci sequence!

Other Interesting Places

Financial Markets: Traders sometimes use "Fibonacci retracement levels" in technical analysis to predict price movements.
Art and Music: Some artists and musicians have used the Fibonacci sequence in their creations, believing it creates a natural harmony.
Conversions: The conversion factor from miles to kilometers (about 1.609) is very close to the golden ratio. This means you can use Fibonacci numbers to roughly convert distances!