Fibonacci number in python

What is the Fibonacci Number Sequence?

The Fibonacci Sequence, as defined by Google, is a collection of numbers.

It is composed of Fibonacci numbers, each of which is the sum of the two previous numbers. The simplest are the sequences 1, 1, 2, 3, 5, 8, and so on.

The Fibonacci Sequence is a mathematical pattern consisting of the following numerics:

The subsequent number is calculated by multiplying the two previous ones.

  • The 2 is calculated by adding the two preceding numbers (1+1).
  • The 3 is obtained by adding the two preceding numbers (1+2), and the 5 is found by adding the two preceding numbers (2+3), and so on!
  • It’s that simple!

Here is a more comprehensive list:

0, 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, etc.

Are you able to deduce the next several numbers?

History

As Parmanand Singh pointed out in 1985, the Fibonacci sequence arises in Indian mathematics in relation to Sanskrit prosody. There was an interest in the Sanskrit poetry tradition in cataloging all patterns of long (L) syllables of two units length contrasted with short (S) syllables of one unit duration. Counting the various patterns of consecutive L and S with a given total duration yields the Fibonacci numbers: the number of patterns with a duration of m units equals Fm + 1.

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As earlier as Pingala (about 450–200 BC), knowledge of the Fibonacci sequence was articulated. Singh quotes Pingala’s enigmatic expression misrau cha («the two are mingled»), which academics understand to mean that the number of sequences forms beats (Fm+1) is achieved by adding one [S] to the Fm cases and one [L] to the Fm1 instances. In the Natya Shastra (c. 100 BC–c. 350 AD), Bharata Muni also indicates an awareness of the sequence. However, the most succinct explanation of the sequence is in the work of Virahanka (about 700 AD), whose original work is lost but whose quote by Gopala is extant (c. 1135).

Regarding Fibonacci The Man

fibonacci

Leonardo Pisano Bogollo was his true name, and he lived in Italy between 1170 and 1250.

His nickname was «Fibonacci,» which translates approximately as «Son of Bonacci.»

Apart from his fame for the Fibonacci Sequence, he was instrumental in the dissemination of Hindu-Arabic Numerals (similar to our modern numerals 0,1,2,3,4,5,6,7,8,9) across Europe in lieu of Roman Numerals (I, II, III, IV, V, etc). That has averted a great deal of hardship for us all! Many thanks, Leonardo.

Fibonacci Day

November 23rd is Fibonacci Day since it contains the numbers «1, 1, 2, 3» that are part of the series. Therefore, on November 23, inform everyone!

The Fibonacci Sequence Rule

The Fibonacci Sequence may be expressed mathematically as a «Rule.»

To begin, the terms are numbered sequentially from 0 to 1 as follows:

table

As a result, the word x6 is used to refer to the sixth term (which equals 8).

As a result, we may write the rule as follows:

The Rule is as follows: xn = xn-1 + xn-2.

  • Here, xn denotes the phrase «n.»
  • The preceding term is xn-1 (n-1)
  • And xn-2 is the preceding term (n-2)

Iterative Python Program for Fibonacci Sequences

This strategy is based on the algorithm described below.

1. Declare two variables to represent the series’s first and second terms. They should be initialized to 0 and 1 as the series’s first and second terms, respectively.
2. Set the value of a variable representing the loop counter to zero.
3. Loop from 0 to the series’s entire number of words.
4. add the variables specified in step 1 to each iteration. This is a word (or item) from the Fibonacci sequence.
B. Assign the second variable’s value to the first and the total from step A to the second variable.
Thus, the following Python program generates a Fibonacci sequence using the preceding approach.

def fibonacci(num): num1 = 0 num2 = 1 series = 0 for i in range(num): print(series, end=' '); num1 = num2; num2 = series; series = num1 + num2; # running function after takking user input num = int(input('Enter how many numbers needed in Fibonacci series- ')) fibonacci(num)

Thus, the Execution’s Output is as follows:

Enter the required number of Fibonacci numbers —

Python Program to Calculate the Fibonacci Sequence Using Recursion

Create a recursive function that takes one input, an integer. This integer input represents the place in the Fibonacci sequence and returns its value. Thus, if it is given 5, it returns the value associated with the fifth place in the Fibonacci sequence.
This recursive function returns 0 or 1 depending on the value of the input. It calls itself with the sum of the nth and (n-1)the places for all other values.

The application retrieves the total number of Fibonacci series components from the keyboard. It then begins a loop from 0 to this input value. Each iteration calls the recursive function and prints the generated Fibonacci item for that place.

def fibonacci(number): # return 0 and 1 for first and second terms if number == 0: return 0 elif number == 1: return 1 else: # return the sum of two numbers return fibonacci(number - 1) + fibonacci(number - 2) # read the total number of items in Fibonacci series max_item_input = input("Enter the number of items in Fibonacci series\n") max_item = int(max_item_input) # iterate from 0 till number of terms for count in range(max_item): print(fibonacci(count), end=",") 

As a result, the output of the preceding operation is

Enter the number of items in Fibonacci series

Applications of the Fibonacci Sequence / Number / Series

  • To begin, the Fibonacci numbers are significant in the computational run-time analysis of Euclid’s method for deciding the greatest common divisor of two integers: the algorithm’s worst-case input is a pair of successive Fibonacci numbers.
  • Additionally, the Fibonacci sequence is an example of a complete series. Thus, any positive integer may be expressed as a sum of Fibonacci numbers, with each number appearing only once.
  • Certain pseudorandom number generators make use of Fibonacci numbers.
  • They are also used in planning poker, a phase in the estimation process for Scrum-based software development projects.
  • Additionally, Fibonacci numbers come into play while analyzing the Fibonacci heap data structure.
  • Fibonacci retracement is a frequently utilized technique in technical analysis for financial market trading.

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Fibonacci series in Python and Fibonacci Number Program

Fibonacci Series in Python

in which each number ( Fibonacci number ) is the sum of the two preceding numbers. The simplest is the series 1, 1, 2, 3, 5, 8, etc.

The Fibonacci Sequence is the series of numbers:

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!

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?

History

The Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1985. [8] [10] [11] In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm + 1. [9]

Knowledge of the Fibonacci sequence was expressed as early as Pingala ( c. 450 BC–200 BC). Singh cites Pingala’s cryptic formula misrau cha (“the two are mixed”) and scholars who interpret it in context as saying that the number of patterns for m beats (Fm+1) is obtained by adding one [S] to the Fm cases and one [L] to the Fm−1 cases. Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD). However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135)

Leonardo Pisano Bogollo Fibonacci Man

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 Hindu-Arabic 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!

The Rule for Fibonacci Series

The Fibonacci Sequence can be written as a “Rule”

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
xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377

So term number 6 is called x6 (which equals 8).

Example: the 8th term is
the 7th term plus the 6th term:
x8 = x7 + x6

The Rule is xn = xn-1 + xn-2

Python Program for Fibonacci Series/ Sequence

Python Program for Fibonacci Series using Iterative Approach

This approach is based on the following algorithm
1. Declare two variables representing two terms of the series. Initialize them to 0 and 1 as the first and second terms of the series respectively.
2. Initialize a variable representing loop counter to 0.
3. Loop from 0 to the total number of terms in the series.
4. In every iteration,
A. add the variables defined in step 1. This represents a term(or item) of the Fibonacci series.
B. assign the value of the second variable to first and the sum in above step A to the second variable.
So Python program to generate Fibonacci series written as per the above algorithm follows.

def fibonacci(num): num1 = 0 num2 = 1 series = 0 for i in range(num): print(series, end=' '); num1 = num2; num2 = series; series = num1 + num2; # running function after takking user input num = int(input('Enter how many numbers needed in Fibonacci series- ')) fibonacci(num)

Thus the Output of the Execution is:

Enter how many numbers needed in Fibonacci series
6
0,1,1,2,3,5,

Python Program for Fibonacci Series using recursion

Create a recursive function which receives an integer as an argument. This integer argument represents the position in Fibonacci series and returns the value at that position. Thus, if it receives 5, it returns the value at 5th position in Fibonacci series.
This recursive function returns 0 and 1 if the argument value is 0 or 1. For all other values, it calls itself with the sum of nth and (n-1)th positions.
The program reads the total number of elements in Fibonacci series from the keyboard. It then initiates a loop starting from 0 till this input value. In every iteration, the recursive function is called and the resultant Fibonacci item for that position is printed.

def fibonacci(number): # return 0 and 1 for first and second terms if number == 0: return 0 elif number == 1: return 1 else: # return the sum of two numbers return fibonacci(number - 1) + fibonacci(number - 2) # read the total number of items in Fibonacci series max_item_input = input("Enter the number of items in Fibonacci series\n") max_item = int(max_item_input) # iterate from 0 till number of terms for count in range(max_item): print(fibonacci(count), end=",")

Thus the output of the above execution is

Enter the number of items in Fibonacci series 8 0,1,1,2,3,5,8,13,

Applications of Fibonacci Series / Sequence / Number

  • First of all the Fibonacci numbers are important in the computational run-time analysis of Euclid’s algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.
  • The Fibonacci numbers are also an example of a complete sequence. So, this means that every positive integer can be written as a sum of Fibonacci numbers, where anyone number is used once at most.
  • Fibonacci numbers are used by some pseudorandom number generators.
  • They are also used in planning poker, which is a step in estimating software development projects that use the Scrum methodology.
  • Also, Fibonacci numbers arise in the analysis of the Fibonacci heap data structure.
  • Retracement of Fibonacci levels is widely used in technical analysis for financial market trading.

So, I hope you liked this article and if you have any questions/recommendations or just want to say hi, comment below!

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