Fibonacci Series | Programming and Data Structures - Computer Science Engineering (CSE) PDF Download

Introduction

The Fibonacci series is a sequence of numbers in which each number (after the first two) is the sum of the two preceding ones. The sequence usually starts with 0 and 1.

The formula to calculate the nth Fibonacci number is:
F(n) = F(n - 1) _ F(n - 2)
Where initial conditions are:
F(0) = 0
F(1) = 1

Here is how the series progresses:
0,1,1,2,3,5,8,13,21,34..

Generating the Fibonacci Series

The Fibonacci series can be generated using various methods, such as recursion, iteration, and dynamic programming.
Below are examples in Python for each method:

Python

Iterative approach:

def fibonacci_iterative(n):

    fib_sequence = [0, 1]

    while len(fib_sequence) <= n:

        fib_sequence.append(fib_sequence[-1] + fib_sequence[-2])

    return fib_sequence[:n+1]

n = 10

print(fibonacci_iterative(n))

Recursive Approach:

def fibonacci_recursive(n):

    if n <= 0:

        return 0

    elif n == 1:

        return 1

    else:

        return fibonacci_recursive(n-1) + fibonacci_recursive(n-2)

n = 10

fib_sequence = [fibonacci_recursive(i) for i in range(n)]

print(fib_sequence)

C++

Iterative Approach:

#include <iostream>

#include <vector>

std::vector<int> fibonacci_iterative(int n) {

    std::vector<int> fib_sequence = {0, 1};

    for (int i = 2; i <= n; ++i) {

        fib_sequence.push_back(fib_sequence[i-1] + fib_sequence[i-2]);

    }

    return fib_sequence;

}

int main() {

    int n = 10;

    std::vector<int> result = fibonacci_iterative(n);

    for (int num : result) {

        std::cout << num << " ";

    }

    return 0;

}

Recursive Approach:

#include <iostream>

int fibonacci_recursive(int n) {

    if (n <= 0) {

        return 0;

    } else if (n == 1) {

        return 1;

    } else {

        return fibonacci_recursive(n-1) + fibonacci_recursive(n-2);

    }

}

int main() {

    int n = 10;

    for (int i = 0; i < n; ++i) {

        std::cout << fibonacci_recursive(i) << " ";

    }

    return 0;

}

Applications of the Fibonacci Series

• Computer Algorithms: Fibonacci sequences are used in algorithms for sorting and searching.
• Mathematics: Fibonacci numbers have applications in number theory, combinatorics, and financial mathematics.
• Nature: Fibonacci patterns appear in biological settings, such as the arrangement of leaves on a stem, the flowering of artichokes, and the arrangement of a pine cone's bracts.
• Art and Architecture: The Fibonacci sequence is used to create visually appealing proportions in art and design.