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:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …..

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.

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

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:

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.

where:

• 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 -

6

0,1,1,2,3,5

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

8

0,1,1,2,3,5,8,13,

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.