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What Is the Fibonacci Sequence?
The Fibonacci sequence is one of the most famous formulas in mathematics.
Each number in the sequence is the sum of the two numbers that precede it. So, the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical equation describing it is Xn+2= Xn+1 + Xn
A mainstay of highschool and undergraduate classes, it’s been called “nature’s secret code,” and “nature’s universal rule.” It is said to govern the dimensions of everything from the Great Pyramid at Giza, to the iconic seashell that likely graced the cover of your school math textbook.
And odds are, almost everything you know about it is wrong.
Scattered history
So then, what’s the real story behind this famous sequence?
Many sources claim it was first discovered or “invented” by Leonardo Fibonacci. The Italian mathematician, who was born around A.D. 1170, was originally known as Leonardo of Pisa, said Keith Devlin, a mathematician at Stanford University. Only in the 19th century did historians come up with the nickname Fibonacci (roughly meaning, “son of the Bonacci clan”), to distinguish the mathematician from another famous Leonardo of Pisa, Devlin said. [Large Numbers that Define the Universe]
But Leonardo of Pisa did not actually discover the sequence, said Devlin, who is also the author of “Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World,” (Princeton University Press, 2020). Ancient Sanskrit texts that used the HinduArabic numeral system first mention it, and those predate Leonardo of Pisa by centuries.
“It’s been around forever,” Devlin told Live Science.
However, in 1202 Leonardo of Pisa published the massive tome “Liber Abaci,” a mathematics “cookbook for how to do calculations,” Devlin said. Written for tradesmen, “Liber Abaci” laid out HinduArabic arithmetic useful for tracking profits, losses, remaining loan balances and so on, Devlin said.
In one place in the book, Leonardo of Pisa introduces the sequence with a problem involving rabbits. The problem goes as follows: Start with a male and a female rabbit. After a month, they mature and produce a litter with another male and female rabbit. A month later, those rabbits reproduce and out comes — you guessed it — another male and female, who also can mate after a month. (Ignore the wildly improbable biology here.) After a year, how many rabbits would you have? The answer, it turns out, is 144 — and the formula used to get to that answer is what’s now known as the Fibonacci sequence. [The 11 Most Beautiful Mathematical Equations]

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“Liber Abaci” first introduced the sequence to the Western world. But after a few scant paragraphs on breeding rabbits, Leonardo of Pisa never mentioned the sequence again. In fact, it was mostly forgotten until the 19th century, when mathematicians worked out more about the sequence’s mathematical properties. In 1877, French mathematician Édouard Lucas officially named the rabbit problem “the Fibonacci sequence,” Devlin said.
The Fibonacci sequence and golden ratio are eloquent equations but aren’t as magical as they may seem. (Image credit: Shutterstock)
Imaginary meaning
But what exactly is the significance of the Fibonacci sequence? Other than being a neat teaching tool, it shows up in a few places in nature. However, it’s not some secret code that governs the architecture of the universe, Devlin said.
It’s true that the Fibonacci sequence is tightly connected to what’s now known as the golden ratio (which is not even a true ratio because it’s an irrational number). Simply put, the ratio of the numbers in the sequence, as the sequence goes to infinity, approaches the golden ratio, which is 1.6180339887498948482. From there, mathematicians can calculate what’s called the golden spiral, or a logarithmic spiral whose growth factor equals the golden ratio. [The 9 Most Massive Numbers in Existence]
The golden ratio does seem to capture some types of plant growth, Devlin said. For instance, the spiral arrangement of leaves or petals on some plants follows the golden ratio. Pinecones exhibit a golden spiral, as do the seeds in a sunflower, according to “Phyllotaxis: A Systemic Study in Plant Morphogenesis” (Cambridge University Press, 1994). But there are just as many plants that do not follow this rule.
“It’s not ‘God’s only rule’ for growing things, let’s put it that way,” Devlin said.
And perhaps the most famous example of all, the seashell known as the nautilus, does not in fact grow new cells according to the Fibonacci sequence, he said.
When people start to draw connections to the human body, art and architecture, links to the Fibonacci sequence go from tenuous to downright fictional.
“It would take a large book to document all the misinformation about the golden ratio, much of which is simply the repetition of the same errors by different authors,” George Markowsky, a mathematician who was then at the University of Maine, wrote in a 1992 paper in the College Mathematics Journal.
Much of this misinformation can be attributed to an 1855 book by the German psychologist Adolf Zeising. Zeising claimed the proportions of the human body were based on the golden ratio. The golden ratio sprouted “golden rectangles,” “golden triangles” and all sorts of theories about where these iconic dimensions crop up. Since then, people have said the golden ratio can be found in the dimensions of the Pyramid at Giza, the Parthenon, Leonardo da Vinci’s “Vitruvian Man” and a bevy of Renaissance buildings. Overarching claims about the ratio being “uniquely pleasing” to the human eye have been stated uncritically, Devlin said.
All these claims, when they’re tested, are measurably false, Devlin said.
“We’re good pattern recognizers. We can see a pattern regardless of whether it’s there or not,” Devlin said. “It’s all just wishful thinking.”
Fibonacci sequence
The Fibonacci sequence is a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum of the preceding two numbers. If the Fibonacci sequence is denoted F (n), where n is the first term in the sequence, the following equation obtains for n = 0, where the first two terms are defined as 0 and 1 by convention:
F (0) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 .
In some texts, it is customary to use n = 1. In that case, the first two terms are defined as 1 and 1 by default, and therefore:
F (1) = 1, 1, 2, 3, 5, 8, 13, 21, 34 .
The Fibonacci sequence is named for Leonardo Pisano (also known as Leonardo Pisano or Fibonacci), an Italian mathematician who lived from 1170 – 1250. Fibonacci used the arithmetic series to illustrate a problem based on a pair of breeding rabbits:
“How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?” The result can be expressed numerically as: 1, 1, 2, 3, 5, 8, 13, 21, 34 .
Fibonacci numbers are of interest to biologists and physicists because they are frequently observed in various natural objects and phenomena. The branching patterns in trees and leaves, for example, and the distribution of seeds in a raspberry are based on Fibonacci numbers.
A Sanskrit grammarian, Pingala, is credited with the first mention of the sequence of numbers, sometime between the fifth century B.C. and the second or third century A.D. Since Fibonacci introduced the series to Western civilization, it has had a high profile from time to time. In The Da Vinci Code, for example, the Fibonacci sequence is part of an important clue. Another application, the Fibonacci poem, is a verse in which the progression of syllable numbers per line follows Fibonacci’s pattern.
The Fibonacci sequence is related to the golden ratio, a proportion (roughly 1:1.6) that occurs frequently throughout the natural world and is applied across many areas of human endeavor. Both the Fibonacci sequence and the golden ratio are used to guide design for architecture, websites and user interfaces, among other things.
Fibonacci Sequence
The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .
The next number is found by adding up the two numbers before it.
 The 2 is found by adding the two numbers before it (1+1)
 The 3 is found by adding the two numbers before it (1+2),
 And the 5 is (2+3),
 and so on!
Example: the next number in the sequence above is 21+34 = 55
It is that simple!
Here is a longer list:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, .
Can you figure out the next few numbers?
Makes A Spiral
When we make squares with those widths, we get a nice spiral:
Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.
The Rule
The Fibonacci Sequence can be written as a “Rule” (see Sequences and Series).
First, the terms are numbered from 0 onwards like this:
n =  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  . 
x_{n} =  0  1  1  2  3  5  8  13  21  34  55  89  144  233  377  . 
So term number 6 is called x_{6} (which equals 8).
Example: the 8th term is
the 7th term plus the 6th term:
So we can write the rule:
Example: term 9 is calculated like this:
Golden Ratio
And here is a surprise. When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio “φ” which is approximately 1.618034.
In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:
Note: this also works when we pick two random whole numbers to begin the sequence, such as 192 and 16 (we get the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, . ):
It takes longer to get good values, but it shows that not just the Fibonacci Sequence can do this!
Using The Golden Ratio to Calculate Fibonacci Numbers
And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:
The answer always comes out as a whole number, exactly equal to the addition of the previous two terms.
When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033. A more accurate calculation would be closer to 8.
Try it for yourself!
You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):
Example: 8 × φ = 8 × 1.618034. = 12.94427. = 13 (rounded)
Some Interesting Things
Here is the Fibonacci sequence again:
n =  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  . 
x_{n} =  0  1  1  2  3  5  8  13  21  34  55  89  144  233  377  610  . 
There is an interesting pattern:
 Look at the number x_{3} = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, . )
 Look at the number x_{4} = 3. Every 4th number is a multiple of 3 (3, 21, 144, . )
 Look at the number x_{5} = 5. Every 5th number is a multiple of 5 (5, 55, 610, . )
And so on (every nth number is a multiple of x_{n}).
1/89 = 0.011235955056179775.
Notice the first few digits (0,1,1,2,3,5) are the Fibonacci sequence?
In a way they all are, except multiple digit numbers (13, 21, etc) overlap, like this:
0.0 
0.01 
0.001 
0.0002 
0.00003 
0.000005 
0.0000008 
0.00000013 
0.000000021 
. etc . 
0.011235955056179775. = 1/89 
Terms Below Zero
The sequence works below zero also, like this:
n =  .  6  5  4  3  2  1  0  1  2  3  4  5  6  . 
x_{n} =  .  8  5  3  2  1  1  0  1  1  2  3  5  8  . 
(Prove to yourself that each number is found by adding up the two numbers before it!)
In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a ++ . pattern. It can be written like this:
Which says that term “n” is equal to (в€’1) n+1 times term “n”, and the value (в€’1) n+1 neatly makes the correct 1,1,1,1. pattern.
History
Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!
About Fibonacci The Man
His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.
“Fibonacci” was his nickname, which roughly means “Son of Bonacci”.
As well as being famous for the Fibonacci Sequence, he helped spread HinduArabic Numerals (like our present numbers 0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.
Fibonacci Day
Fibonacci Day is November 23rd, as it has the digits “1, 1, 2, 3” which is part of the sequence. So next Nov 23 let everyone know!
What is the Fibonacci Sequence (aka Fibonacci Series)?
Leonardo Fibonacci discovered the sequence which converges on phi.
In the 1202 AD, Leonardo Fibonacci wrote in his book “Liber Abaci” of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind phi. This sequence was known as early as the 6th century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa. He is also known as Leonardo Bonacci, as his name is derived in Italian from words meaning “son of (the) Bonacci”.
Starting with 0 and 1, each new number in the sequence is simply the sum of the two before it.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 . . .
This sequence is shown in the right margin of a page in Liber Abaci, where a copy of the book is held by the Biblioteca Nazionale di Firenze. Click to enlarge.
The relationship of the Fibonacci sequence to the golden ratio is this: The ratio of each successive pair of numbers in the sequence approximates Phi (1.618. . .) , as 5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60.
The table below shows how the ratios of the successive numbers in the Fibonacci sequence quickly converge on Phi. After the 40th number in the sequence, the ratio is accurate to 15 decimal places.
1.618033988749895 . . .
Compute any number in the Fibonacci Sequence easily!
Here are two ways you can use phi to compute the nth number in the Fibonacci sequence (f_{n}).
If you consider 0 in the Fibonacci sequence to correspond to n = 0, use this formula:
Perhaps a better way is to consider 0 in the Fibonacci sequence to correspond to the 1st Fibonacci number where n = 1 for 0. Then you can use this formula, discovered and contributed by Jordan Malachi Dant in April 2005:
f_{n} = Phi n / (Phi + 2)
Both approaches represent limits which always round to the correct Fibonacci number and approach the actual Fibonacci number as n increases.
The ratio of successive Fibonacci numbers converges on phi
Sequence in the sequence 
Resulting Fibonacci number (the sum of the two numbers before it) 
Ratio of each number to the one before it (this estimates phi) 
Difference from Phi 
1  1  
2  1  1.000000000000000  +0.618033988749895 
3  2  2.000000000000000  0.381966011250105 
4  3  1.500000000000000  +0.118033988749895 
5  5  1.666666666666667  0.048632677916772 
6  8  1.600000000000000  +0.018033988749895 
7  13  1.625000000000000  0.006966011250105 
8  21  1.615384615384615  +0.002649373365279 
9  34  1.619047619047619  0.001013630297724 
10  55  1.617647058823529  +0.000386929926365 
11  89  1.618181818181818  0.000147829431923 
12  144  1.617977528089888  +0.000056460660007 
13  233  1.618055555555556  0.000021566805661 
14  377  1.618025751072961  +0.000008237676933 
15  610  1.618037135278515  0.000003146528620 
16  987  1.618032786885246  +0.000001202064649 
17  1,597  1.618034447821682  0.000000459071787 
18  2,584  1.618033813400125  +0.000000175349770 
19  4,181  1.618034055727554  0.000000066977659 
20  6,765  1.618033963166707  +0.000000025583188 
21  10,946  1.618033998521803  0.000000009771909 
22  17,711  1.618033985017358  +0.000000003732537 
23  28,657  1.618033990175597  0.000000001425702 
24  46,368  1.618033988205325  +0.000000000544570 
25  75,025  1.618033988957902  0.000000000208007 
26  121,393  1.618033988670443  +0.000000000079452 
27  196,418  1.618033988780243  0.000000000030348 
28  317,811  1.618033988738303  +0.000000000011592 
29  514,229  1.618033988754323  0.000000000004428 
30  832,040  1.618033988748204  +0.000000000001691 
31  1,346,269  1.618033988750541  0.000000000000646 
32  2,178,309  1.618033988749648  +0.000000000000247 
33  3,524,578  1.618033988749989  0.000000000000094 
34  5,702,887  1.618033988749859  +0.000000000000036 
35  9,227,465  1.618033988749909  0.000000000000014 
36  14,930,352  1.618033988749890  +0.000000000000005 
37  24,157,817  1.618033988749897  0.000000000000002 
38  39,088,169  1.618033988749894  +0.000000000000001 
39  63,245,986  1.618033988749895  0.000000000000000 
40  102,334,155  1.618033988749895  +0.000000000000000 
Tawfik Mohammed notes that 13, considered by some to be an unlucky number, is found at position number 7, the lucky number!
The Fibonacci Sequence and Gambling or Lotteries
Some people hope that Fibonacci numbers will provide an edge in picking lottery numbers or bets in gambling. The truth is that the outcomes of games of chance are determined by random outcomes and have no special connection to Fibonacci numbers.
There are, however, betting systems used to manage the way bets are placed, and the Fibonacci system based on the Fibonacci sequence is a variation on the Martingale progression. In this system, often used for casino and online roulette, the pattern of bets placed follows a Fibonacci progression: i.e., each wager should be the sum of the previous two wagers until a win is made. If a number wins, the bet goes back two numbers in the sequence because their sum was equal to the previous bet.
In the Fibonacci system the bets stay lower then a Martingale Progression, which doubles up every time. The downside is that in the Fibonacci roulette system the bet does not cover all of the losses in a bad streak.
An important caution: Betting systems do not alter the fundamental odds of a game, which are always in favor of the casino or the lottery. They may just be useful in making the playing of bets more methodical, as illustrated in the example below:
The Fibonacci sequence
The Fibonacci sequence has been named after Leonardo of Pisa also known as Fibonacci (a mix of the words Filius Bonacci, which means son of Bonacci). He first described this sequence in the year 1202 in his book Liber Abaci. Although he is seen as the first who discovered this sequence, It was later discovered that this sequence was already known by Indian mathematicians.
The Fibonacci sequence:
0, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 …
Each element in the sequence comes by adding the last two elements. For instance, the number 13 is achieved by adding the numbers 5 and 8 and the number 21 is achieved by adding 8 with 13. You might not help wondering why is the sequence so famous? What is Fibonacci? The explanation can be seen if the sequence is depicted visually since then it becomes clear that the sequences describes a growth pattern in nature. See the picture below which explains the fibonacci spiral.
The number 1 in the sequence stands for a square with each side 1 long. The number 2 stands for a square of 2 by 2 and so on. If the sides of the square are placed next to each other a new side of a larger square forms as explained before, e.g. 2+3 gives 5 and same goes for the squares. This can be repeated till infinity and with each step the squares get larger. Thing that is so special about this sequence can be seen when a line is drawn trough the cross points.
With these picture is becomes clear what the sequence actually represents. This pattern is seen in many natural phenomenon, for example in the smallest nautilus and even in the shape of the largest galaxy’s. The sequence also has directly connected with the golden ratio and is used throughout history in many works of art such as the Mona Lisa, but it doesn’t stop here, the Fibonanci sequence can even be heard in music. The best example is Lateralus by metal band Tool.

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