The Fibonacci sequence is a series of numbers in which each number is the sum of the two before it. It starts with 0 and 1, so it runs 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and on forever. The simple rule behind the Fibonacci sequence produces a pattern that turns up in mathematics, computer science and the way many plants grow.
The sequence is named after Leonardo of Pisa, known as Fibonacci, who described it in his book Liber Abaci in 1202. Indian mathematicians had worked with the same numbers much earlier, when counting the rhythm patterns of Sanskrit poetry.
Key facts about the Fibonacci sequence
- Rule: F(n) = F(n-1) + F(n-2), starting from F(0) = 0 and F(1) = 1
- First terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
- Named after: Leonardo of Pisa (Fibonacci), Liber Abaci, 1202
- Earliest known roots: Pingala, Indian mathematics, roughly 450 to 200 BC
- Name coined by: the 19th-century French mathematician Édouard Lucas
- Golden ratio link: the ratio of neighbouring terms approaches about 1.618
- Pattern: every third Fibonacci number is even
Source: Fibonacci sequence, Wikipedia.
Historical context
Indian origins
Knowledge of the numbers goes back to Pingala, who studied how many ways long and short syllables can fill a line of verse. Counting those patterns gives exactly the Fibonacci numbers.
Leonardo Pisano and the Liber Abaci
Leonardo Pisano, better known as Fibonacci, brought the sequence to Europe in his 1202 book Liber Abaci. The book is best known for introducing Hindu-Arabic numerals to European merchants, which he had learned about during his travels in North Africa.
In the book the sequence appears as a puzzle about rabbits. A pair of rabbits produces a new pair every month, and each new pair starts breeding after a month. The number of pairs month by month follows the Fibonacci sequence.
How the sequence got its name
Mathematicians kept returning to the sequence for centuries. Johannes Kepler noticed that the ratio of neighbouring terms settles toward a fixed value. The name “Fibonacci sequence” itself was first used by Édouard Lucas in the 19th century.
Mathematical definition
What is the Fibonacci sequence?
The Fibonacci sequence starts with 0 and 1. Each new number is the sum of the previous two:
- 0 + 1 = 1
- 1 + 1 = 2
- 1 + 2 = 3
- 2 + 3 = 5
- 3 + 5 = 8, and so on
Written as a recursive formula, the rule is F(n) = F(n-1) + F(n-2). Here F(n) is any term, F(n-1) is the term just before it and F(n-2) is the one before that.
The first 20 Fibonacci numbers
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181.
Properties of the Fibonacci sequence
Unique characteristics
The numbers grow quickly, roughly multiplying by 1.618 at every step. Every third number is even, because odd plus odd gives even and the pattern odd, odd, even repeats.
The sequence also hides in Pascal’s Triangle: add up the numbers along its shallow diagonals and you get the Fibonacci numbers.
Connection with the golden ratio
The golden ratio, written phi (φ), is about 1.618. Divide a Fibonacci number by the one before it and you get close to phi, and the bigger the numbers, the closer the result. For example, 89 divided by 55 is about 1.618.
The golden spiral is drawn by joining quarter circles inside squares whose sides are Fibonacci numbers.
Some popular claims go too far. Careful measurements do not support the idea that the Parthenon or the Mona Lisa were designed around the golden ratio, and nautilus shells do not grow by it either, according to the golden ratio article on Wikipedia.
Calculation methods
Binet’s formula
Binet’s formula calculates any Fibonacci number directly from the golden ratio:
F(n) = (φ^n – (1-φ)^n) / √5
Here φ is the golden ratio and n is the position in the sequence. The recurrence F(n) = F(n-1) + F(n-2) is easier to understand, but you have to work through every earlier term to use it.
Using recursion in code
Recursion is a popular way to calculate Fibonacci numbers in programming. A basic Python version:
def fibonacci(n):
if n <= 1:
return n
return fibonacci(n-1) + fibonacci(n-2)
This version recalculates the same values many times and becomes slow for large n. A simple loop, or storing results as you go (memoization), makes it fast.
The Fibonacci sequence in nature
Fibonacci numbers appear in many plants, though not in all species. In 1830 the botanists Karl Friedrich Schimper and Alexander Braun found that the spiral arrangements of leaves and scales often follow Fibonacci numbers.
The seed heads of sunflowers are the best-known example, with spirals in one direction and a neighbouring Fibonacci number of spirals in the other. Pine cones, pineapples, artichokes and the branching of trees show the same kind of pattern. The family tree of a male honeybee also follows the sequence.
These arrangements let a plant pack seeds or leaves evenly, without large gaps or overlaps.
Fibonacci in art and design
Artists and designers often use the Fibonacci spiral or golden rectangles as a guide for placing key elements in an image, and some composers have used Fibonacci numbers to structure their music. Fibonacci in art is mostly a modern design habit, and many famous historical examples are disputed.
Fibonacci in finance
Traders use Fibonacci retracements on price charts. They mark levels such as 38.2%, 50% and 61.8% of a price move and watch whether the price stalls there. Fibonacci extensions are used in the same way to set price targets.
No method predicts the market reliably, and Fibonacci levels are a charting habit rather than a law of economics.
Fibonacci in popular culture
Dan Brown’s novel The Da Vinci Code uses the sequence as part of a code. Poets write “Fibonacci poems” whose lines have 1, 1, 2, 3, 5 and 8 syllables. Mathematicians even run a journal devoted to the numbers, the Fibonacci Quarterly.
Related sequences
The Lucas numbers follow the same rule but start with 2 and 1, giving 2, 1, 3, 4, 7, 11, 18. Fibonacci numbers also appear in continued fractions for the golden ratio, and in combinatorics they count the ways to write a number as a sum of 1s and 2s.
Frequently asked questions
What is the Fibonacci sequence in simple terms?
It is a list of numbers where each number is the sum of the two before it, starting with 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13 and so on.
What is the formula for the Fibonacci sequence?
The rule is F(n) = F(n-1) + F(n-2), with F(0) = 0 and F(1) = 1. Binet’s formula, F(n) = (φ^n – (1-φ)^n) / √5, gives any term directly using the golden ratio.
Who discovered the Fibonacci sequence?
Indian mathematicians such as Pingala described the numbers centuries before Fibonacci. Leonardo of Pisa (Fibonacci) introduced the sequence to Europe in Liber Abaci in 1202, and Édouard Lucas gave it its name in the 19th century.
How is the Fibonacci sequence related to the golden ratio?
Dividing a Fibonacci number by the one before it gives a value that gets closer and closer to the golden ratio, about 1.618, as the numbers grow.
Where does the Fibonacci sequence appear in nature?
In the spirals of sunflower seed heads, pine cones, pineapples and artichokes, in the arrangement of leaves on many stems, and in the family tree of honeybees.