Serial number of a number of Fibonacci
 Serial numbers and values of the next numbers of Fibonacci

For the first time, Fibonacci numbers appeared in the work "Liber abacci" ("The book about the abacus"), written by the famous Italian mathematician Leonardo from Pisa, who is better known by his nickname Fibonacci (Fibonacci - abbreviated filius Bonacci, i.e., son of Bonachchi). This book, written in 1202, came to us in its second version, which refers to 1228.

Almost at the time of the dark Middle Ages, this is the only book from Europe dated to the 13th century devoted to mathematics.

Liber abacci is a voluminous work that contains almost all the arithmetic and algebraic information of the time and played a prominent role in the development of mathematics in Western Europe over the next several centuries.

In particular, it was from this book that Europeans became acquainted with Hindu (“Arab”) figures.

The material reported in Liber abacci is explained on a large number of tasks, which constitute a significant part of this treatise.

Fibonacci numbers arise in a wide variety of areas of life.

On a sunflower, the seeds are arranged in spirals, and the number of spirals going in one direction and the other is different - they are consecutive Fibonacci numbers (for example, spirals can be 34 and 55).

The same is observed on the fruits of pineapple, where there are usually 8 spirals.

Fibonacci numbers 3, 5, 8, 13 appear in a curious geometric sophism, claiming that “64 = 65” or “lost area” by cutting the square 8X8 and folding a rectangle 5X13 from it.

Fibonacci numbers also arise when describing a winning strategy in the ancient Chinese game of "dzyanypitszy", in which two players take turns taking stones from two piles: either an arbitrary amount from one pile, or equally from two (the player who takes the last stone wins).

It was found that fractions of the a / c type corresponding to the helical arrangement of leaves on the plant stem are often the ratios of successive Fibonacci numbers.

For beech and hazel, this ratio is 2/3, for oak and apricot - 3/5, for poplar and pear - 5/8, for willow and almond - 8/13, etc.

Fibonacci numbers were inextricably linked with the so-called "golden ratio", which was used and practically erected in a kind of "cult" by the ancient Greeks.

The golden ratio is the division of an arbitrary segment in such a way that   most refers to the smaller, just as the entire segment refers to the larger.

Figures, paintings, buildings and everything that surrounds us, according to the ancients, everything should obey the "golden ratio". The correctness of this ratio indicates the external harmony of the observed object.

If we want to solve the problem and determine in numbers what the golden ratio is, we need to solve the proportion

$\frac{a}{b}=\frac{a+b}{a}$

If we take the smaller part of b as unity, and denote a by x, we get

${x}=\frac{x+1}{x}$

$x^2-x-1=0$

The solution of this equation gives us the following result

$x=\frac{1\pm\sqrt{5}}{2}$

A positive number equal to $x=1.6180339887499$ and there is a golden ratio

The most surprising thing is that despite the obvious irrationality of the resulting number, it is the “golden section” that will help us find an arbitrary number of the Fibonacci series. To do this, we use the so-called Binet formula:

$F_n = \frac{\left(\frac{1 + \sqrt{5}}{2}\right)^n - \left(\frac{1 - \sqrt{5}}{2}\right)^n}{\sqrt{5}}$

This formula is used by the bot to give you an exact answer.

And also about the "golden section". Let's try to translate the number $x=1.6180339887499$ in a chain or in another, continuous fraction. To do this, use the bot Continuous, continued fractions online

We will see that the irrational number $x=\frac{1+\sqrt{5}}{2}$ has a beautiful, in its beauty, continuous fraction.

$\frac{1+\sqrt{5}}{2}=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{...+\cfrac{1}{1}}}}$

## The sum of the Fibonacci series

If we carefully look at the Fibonacci series 1, 1, 2, 3, 5, 8.13, 21.34, we will notice another feature.

The sum of the Fibonacci series is equal to the number of this series increased by two positions minus two.

For example, what is the sum of the first six numbers of this series 1 + 2 + 3 + 5 + 8 + 13?

It is equal to the eighth (six + two) Fibonacci numbers minus 2. that is 34-2 = 32

Thus, in order to calculate the sum of a series, it is not necessary to use cycles or to sum each element, it suffices to find the Fibonacci number, whose number is two more from the given value and subtract two by the Binet formula.

## Syntax

Jabber: fb <number>

WEB: <number>

A number can be an integer that denotes a sequence number in a Fibonacci series.

0,1,1,2,3,5,8 ......

## Examples

Example:

Define the 30th number of the series

fb 30

The 30th number of the Fibonacci series is 1346269
The following numbers in this series are equal
The 31st number is 2178309
32nd number equals 3524578
33rd number is 5702887
34th number is 9227465
The 35th number is 14930352
36th number is 24157817
37th number equals 39088169
38th number equals 63245986
39th number is 102334155

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