Named after the Italian mathematician Leonardo Pisano Bogollo, later known as Fibonacci, the Fibonacci sequence is a series of Fibonacci numbers known as F n 44 44. The Fibonacci series is given as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 38, In the Fibonacci sequence, each term is the sum of the two preceding terms, starting with 0 and 1 as the first and second terms. In some older references, the term “0” can be omitted. The first two terms are “0” and “1” because the Fibonacci sequence is a series of numbers (also known as Fibonacci numbers), where each number is the sum of the two preceding numbers. In some older versions of the series, the term “0” can be omitted. Therefore, the Fibonacci sequence is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. Therefore, we can see that each term can be calculated by adding two terms before it.

## Fibonacci Series Applications

Fibonacci sequences are used in various areas of our daily lives. The different patterns found in different fields, from nature to music to the human body, follow the Fibonacci sequence. Some of the uses of the Fibonacci sequence are given as

•First and foremost, the Fibonacci number is important in the computational run-time analysis of the Euclidean algorithm to determine the greatest common divisor of two integers. The worst input of this algorithm is a pair of consecutive Fibonacci numbers.

•Used for grouping numbers and for studying various other special mathematical sequences.

•Used in coding (computer algorithms, distributed systems, etc.). The Fibonacci sequence is important, for example, in the computational run-time analysis of the Euclidean algorithm used to determine the GCF of two integers.

•Used in many fields of science, such as quantum mechanics and cryptography.

•In financial market transactions, the Fibonacci retracement level is widely used in technical analysis.

•The Fibonacci number is also an example of a complete sequence. This means that positive integers can be described as the sum of Fibonacci numbers, and each number can be used up to once.

•The Fibonacci number is used by some pseudo-random number generators. It is also used for poker planning, which is the step in estimating software development projects that use the Scrum technique.

•The Fibonacci number is also created when analyzing the Fibonacci heap data structure. Fibonacci level retracements are widely used in technical analysis of financial market transactions.

## Fibonacci series in python

The Fibonacci Series is a sequence of integers in which each number is the outcome of adding the two numbers before it. The first two integers are 0 and 1; so, the third number is ought to be the addition of previous 2 numbers that is 0+1 which is 1 and then for the 4th and upcoming sequential numbers the same process is repeated.

Fibonacci sequence of the first 10 numbers:

0,1,1,2,3,5,8,13,21,34….

When we build squares with these widths, we obtain a spiral, which is an intriguing characteristic of these numbers.A Fibonacci spiral is a drawing of quarter circles connected inside a square block with the Fibonacci numbers printed on each block.

## Fibonacci sequence formula

To calculate Fibonacci Series, we need to use the following formula:

Fn = Fn-1 + Fn-2 ; where

Fn is term number “n”

Fn-1 is the previous term i.e. (n-1)

Fn-2 is the previous term of (n-1) i.e. (n-2)

## Fibonacci series in python using for loop

Python For Loop allows you to loop through a series of statements once for each element of a sequence or collection. A sequence or collection can be a range, list, tuple, dictionary, set, or string. In this tutorial you will learn how to implement a for loop in each of the above collections. In the future, we plan to provide a detailed sample program using Python for Loop. When the program execution first enters the for loop, it checks the element from the iterable. If the element is available, the program executes the instruction within the block. After executing the instruction, the program checks if the following elements are available: If True, the instruction for this next element will be executed again. The for-loop block is executed for each element of iterable. After executing an instruction on every element in an iteration, the execution is in a situation where there is no next element in the iteration. Since there is no next element at this point, the for loop is considered complete and execution continues with the next statement in the program. The for loop can also contain an optional else block. The else part is executed when the elements of the sequence used for the loop are exhausted. You can use the break keyword to break the for loop. In such type of cases, the else part is ignored. Therefore, if there is no interruption, the else part of the for loop is executed.

**Algorithm**

1: Declare two variables that represent the two terms in the series. Initialize to 0 and 1, respectively, as the first and second terms of the series.y.

2. Initialize another variable to 0. This will be the for-loop counter.

3. Loop from 0 to the total number of terms in the sequence.

4. In every iteration,

A. add the variables defined in step 1. This represents a term of the Fibonacci sequence**.**

B. Assign the value of the second variable to the first variable and the sum from step A above to the second variable

## EXample

1 2 3 4 5 6 7 8 9 10 11 | terms = int (input ("Enter the Number of terms required:")) num1=0; num2=1; num3=0; print(num1) print(num2) for n in range (2,terms): num3=num1+num2 print(num3) num1=num2 num2=num3 |

1 2 | Enter the Number of terms required: 10 0 1 1 2 3 5 8 13 21 34 55 89 |

## Fibonacci series in python using while loop

Python While Loops are used to iterate over a series of statements based on the output of a Boolean expression. The While Loop is one of Python’s loop statements. Is a Python keyword, a condition is a Boolean expression, and an instruction is a block of code. Statements in a while loop must be indented as shown in the syntax. When using a while loop, a Boolean expression can contain one or more variables. These variables must be initialized before the while loop and updated inside the while loop. The programmer needs to make sure that the condition is broken or failing. Otherwise, this while loop can become an infinite while loop. The programmer needs to make sure that the while loop ends at some point during execution. Otherwise, it can be repeated indefinitely, but this is not always desirable.

**Algorithm:**

Step 1: Input the ‘n’ value

Step 2: Initializing sum = 0, a = 0, b = 1 and count = 1

Step 3: while (count <= n)

Step 4: print sum

Step 5: Increment the count variable

Step 6: swap a and b

Step 7: sum = a + b

Step 8: while (count > n)

Step 9: End the algorithm

Step 10: Else

Step 11: Repeat from step 4 to step 7

1 2 3 4 5 6 7 8 9 10 11 12 13 | terms = int (input ("Enter the Number of terms required:")) num1=0; num2=1; num3=0; term=0; print(num1) print(num2) while term<terms: num3=num1+num2 print(num3) num1=num2 num2=num3 term+=1 |

1 2 | Enter the Number of terms required: 10 0 1 1 2 3 5 8 13 21 34 55 89 |

## Example2

## Fibonacci series in python using list

Python lists are the most powerful way to store different types of items. Python has different types of list functions that allow you to add, remove, sort, and undo list items. This section provides a list of Python list functions with examples. We’ll also add some common Python list functions such as the sum, min, and max functions. Python encloses items in square brackets [] and separates them with commas to create a list. The list can contain any element. The list can also contain another list as an element. This is known as a nested list. It is a number of elements and can be of any type (integer, floating point, string, etc.). The Python list data type has individual functions or methods for adding, deleting, searching, editing, and flipping list items.

**Algorithm:**

**•**First, the user enters any positive integer

•The for loop is used to iterate over a specified number of values.

•Finally print (fib) to add the Fibonacci sequence to the list**.**

## Example3

**Fibonacci series in python using function**

**Algorithm:**

- In this example, we used the function as def fib (n). Initialized n1 to 0 and n2 to 1.
- if n == 1 then print (n1) Uses a for loop to iterate over the specified number of values
- Finally, the Fibonacci sequence is printed

## Example

**Fibonacci series up to n terms**

**Algorithm**

**•**First, I initialized n1 to 0 and n2 to 1 to allow the user to enter n terms.

•If the number of terms exceeds 2, use a while loop to find the next term in the sequence by summing the two preceding terms.

•Then exchange variables to continue the process.

## Example

**Fibonacci series using recursion**

Recursion is the process of defining something from your own point of view. An example of the physical world is placing two parallel mirrors facing each other. Objects in between are reflected recursively. The recursion ends when the number is reduced to 1. This is called the basic condition. All recursion functions require a basic condition to stop recursion. Otherwise, the function calls itself indefinitely. The Python interpreter limits the depth of recursion to avoid infinite recursion leading to stack overflow. The recursive function makes your code look neat and elegant. Complex tasks can be broken down into simpler sub-problems using recursion. Generating a sequence is easier with recursion than with nested iterations.

Examining the Recursion Behind the Fibonacci Sequence

The generation of the Fibonacci sequence is a classical recursive problem. Recursion is when a function calls itself to solve the problem it is trying to solve. In each function call, the problem gets smaller until it hits a base case, then it returns the result to each intermediate caller until it returns the final result to the caller. ‘Origin. You must first calculate the predecessors of the Fibonacci number F (5), F (4) and F (3), if you want to calculate it. And to calculate F (4) and F (3), you will need to compute them first. The decomposition of F (5) into smaller subproblems looks like this**:**

X (5) =X (4) + X (3)

= X (3) + X (2) + X (2) + X (1)

= X (2) + X (1) + X (1) + X (0) + X (1) + X (0) + 1

= X (1) + X (0) + 1 + 1 + 0 + 1 + 0 + 1

= 1 + 0 + 1 + 1 + 0 + 1 + 0 + 1

= 5

Generating the Fibonacci sequence Recursively in Python

Create a recursive function that takes an integer as an argument. This integer argument returns the value at the position of the Fibonacci sequence. So, getting 5 returns the value at the 5th position in the Fibonacci sequence. This recursive function gives the value 0 and 1 if the argument value is 0 or 1. For all other values, call itself with the sum of the nth and (n1) the positions. The program is reading the total number of elements in the Fibonacci sequence from the keyboard. Then it starts a loop from 0 to this input value. At each iteration, a recursive function is called and the resulting Fibonacci element is printed at that location.

Algorithm:

• In this example, we used the def recursion fib (num) function. If num <= 1, returns num.

• The other terms are obtained by adding the previous two terms. Term n is the sum of terms (num1) and (num2).

• Use the recursive function recursion fib to compute the n-term sequence. The for loop is used to recursively iterate over each term.

## Example

## Python program to print Fibonacci series without using recursion

Algorithm:

•First, the user enters the first two numbers in the series and the number of terms the user prints.

•Then print the first two numbers.

• Use the while loop to sum the first two numbers and then output the Fibonacci sequence until the Fibonacci sequence n2 is greater than 0.

## Program to print first 50 Fibonacci numbers in python

Algorithm:

• In this example, we used the function defFibonacciNum (n).

• Initialized n1 to 0 and n2 to 1. For n < ;, 1 is returned and is returned.

• The for loop is used to iterate over a specified number of values.

• Finally, call the FibonacciNum (50) function to get the first 50 Fibonacci numbers in the output

## Fibonacci series using dynamic programming

**Fibonacci series in python using iteration**

Container objects such as lists and sets can contain many elements. A container object is said to be repeatable if you can access the elements of the container object one by one. Our program uses a variety of repeatable objects such as lists, tuples, sets, and dictionaries. Iterators are repeatable objects. That is, you can use an iterator to access all the elements in a repeatable object. In Python, you can use the iter () method to create an iterator for each container object. The iter () method takes a repeatable object as input and returns an iterator for the same object.

**Algorithm:**

This approach is dependent on the following algorithm

- 1: Declare two variables that represent the two terms in the series. Initialize to 0 and 1, respectively, as the first and second terms of the series.
- Initialize the variable that represents the loop counter to 0.
- The third loop from 0 to the total number of terms present in the series.
- In every iteration
- A for each iteration. Add the variables defined in step 1. It represents a term (or element) in the Fibonacci sequence.
- Assign the value of the second variable to the first variable and the sum from step A above to the second variable.