Fibonacci Calculator

The Fibonacci sequence starts with 0 and 1, after which each number is the sum of the two preceding ones. So it continues like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. Expressed as a rule, it looks like this: $F_n = F_{n-1} + F_{n-2}$.

This calculator offers three functions: finding any number in the fibonacci sequence, generating the sequence up to a certain number of elements and checking if a given number is a fibonacci number. By selecting one of the modes above, the result will be shown in the corresponding field.

What is the Fibonacci sequence?

The Fibonacci sequence is defined by two initial values and a rule. The first two terms are $F_0 = 0$ and $F_1 = 1$, where each subsequent term is the sum of the previous two terms, i.e. $F_n = F_{n-1} + F_{n-2}$. Such a definition in which new terms are generated on the basis of previous terms is called a recurrence relation.

To manually calculate a single term, you simply add. To get $F_4$, you add the two previous terms, namely $F_4 = F_3 + F_2 = 2 + 1 = 3$. By continuing to calculate, the numbers grow quickly as each term takes on all of the influences accumulated so far.

This concept has a long history. Centuries ago Indian scholars described this pattern as part of their study of poetry. Later around 1202 Leonardo of Pisa, better known as Fibonacci, introduced the concept to Europe through a problem about rabbit breeding. Although the rabbits are fictitious it turns out that the resulting sequence of numbers forms the basis for an astonishingly large number of phenomena in mathematics and nature.

The calculation of the nth Fibonacci number.

One direct method of computing a particular Fibonacci number is to sequentially build up a table of Fibonacci numbers, starting with the first two values, 0 and 1, and then continuing to add two numbers together as many times as necessary to reach the desired position. The table on this page shows exactly this method, listing the values from F(0) through the desired position.

Another way is to use a closed formula that directly applies to any element. This formula is called the Binet's Formula and consists of the golden ratio $\varphi = \tfrac{1 + \sqrt5}{2}$ and its conjugate value $\psi = \tfrac{1 - \sqrt5}{2}$.

$\psi$ is a small negative number and $\psi^n$ gets smaller as $n$ gets larger. So the result at any position is the whole number closest to $\varphi^n / \sqrt5$. This means that Fibonacci numbers, which are all whole numbers, come from a formula involving the square root of an irrational number.

The golden ratio hidden within the proportions.

When any Fibonacci number is divided by the one before it, the result approximates a constant value known as the Golden Ratio. In the early stages of the sequence, the ratios fluctuate like $\tfrac{2}{1} = 2$, $\tfrac{3}{2} = 1.5$ and $\tfrac{5}{3} \approx 1.667$, but they converge rapidly so that after about a handful of terms, the decimal places are nearly identical. You can observe this shift easily in the ratio column of the table.

This explains why the Fibonacci numbers appear in a wide variety of contexts where the golden ratio appears, from the spirals in sunflower seed heads to the proportions that artists strive for. The golden ratio is precisely the number that the ratios of the Fibonacci numbers approach more and more closely.

The sum of Fibonacci numbers:

The sum of consecutive terms has a simple trick to it. When you add the first few Fibonacci numbers together, their sum is always one less than the Fibonacci number two places over.

One can easily verify this. The individual terms from $F_0$ to $F_6$ are 0, 1, 1, 2, 3, 5, 8, and their sum is 20. In fact it's $F_8 - 1 = 21 - 1 = 20$. The total sum function used on this page uses that identity instead of adding up each individual term.

How to check if a number is a Fibonacci number:

There is a way to generate the fibonacci numbers until you get a number that is larger than the one you are investigating. This is actually what a testing tool does internally. There is also another clever method of determining if a number is a Fibonacci number without having to write out the sequence. A non-negative integer $n$ is a Fibonacci number if and only if either $5n^2 + 4$ or $5n^2 - 4$ is a perfect square.

For example, take 144. It is both $5 \times 144^2 + 4 = 103684$ and $322^2 = 103684$, so it's a perfect square. Consequently, 144 is the fibonacci number at position 12. Let's look at 100. Since neither $5 \times 100^2 + 4 = 50004$ nor $5 \times 100^2 - 4 = 49996$ are perfect squares, 100 is not a fibonacci number. It falls between 89 and 144.

Fibonacci numbers for negative positions

The sequence can be extended to negative position numbers by using the same recursive relationship in reverse, starting from F1 = 1 and F0 = 0. Calculating backwards gives the sequence 1, −1, 2, −3, 5, −8... The absolute value of each term is the corresponding Fibonacci number, with alternating signs following F(-n) = (-1)^(n+1) F(n).

This tool covers positions that are greater than or equal to 0. This is the convention most people refer to as the Fibonacci sequence. The negative side is also shown here so you can see the full picture.

Does the sequence start at 0 or 1?

Both conventions are used, which often leads to confusion. The modern standard followed by this tool starts at zero: $F_0 = 0$, $F_1 = 1$, $F_2 = 1$, $F_3 = 2$. In older books and puzzles the leading zero is sometimes dropped and one is started with, so that in those the nth element corresponds to the $(n{-}1)$th element of this tool. Where position matters, this page will show an index for each element, so it's clear which element is being considered.

Where does the Fibonacci sequence appear?

In plants the number of spirals on a pine cone, pineapple or sunflower seed head is often found to match a Fibonacci number. This is because the arrangement of seeds or leaves in an angle relative to the Golden Ratio helps them fill space most efficiently. The same numbers appear when counting the branching of some trees and the arrangement of leaves around their stems.

In computer science, the sequence is a classic example of recursion and illustrates the value of storing previously computed results. The Fibonacci numbers are also the basis for the name "Fibonacci heap" data structure, used in computer programming. They also appear in search techniques, in the analysis of Euclid's algorithm, and in number theory, where every positive integer can be written as the sum of non-consecutive Fibonacci numbers.

Table of first Fibonacci numbers:

Here are the individual terms from F(0) to F(20). If you switch the calculation tool into list mode, you can extend it as far as you like.