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.
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 socalled "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
If we take the smaller part of b as unity, and denote a by x, we get
The solution of this equation gives us the following result
A positive number equal to 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 socalled Binet formula:
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 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 has a beautiful, in its beauty, continuous fraction.
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 342 = 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
we get the answer
