**What is Fibonacci Series in Java?**

**Fibonacci series** are the numbers in the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21….. The series in the Fibonacci sequence is equal to the sum of the previous two terms. The Fibonacci sequence’s first two terms are **0** and **1** respectively.

Mathematically, we can denote it as:

**F _{n} = F_{n-1} + F_{n-2}**

Where, F_{n} denotes the nth term of Fibonacci series.

The first two terms of this series are considered to be:

F_{0} = 0 (Zeroth term of Fibonacci sequence)

F_{1} = 1 (First term of Fibonacci sequence)

Now, by using the above two values we can easily calculate all other terms of Fibonacci series as follows :

F_{2} = F_{1} + F_{0} = 0 + 1 = 1

F_{3} = F_{2} + F_{1} = 1 + 1 = 2

F_{4} = F_{3} + F_{2} = 2 + 1 = 3

Series is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ……

Write a Java Program to generates fibonacci series.

In fibonacci series the first two numbers in the Fibonacci sequence are 0 and 1 and each subsequent number is the sum of the previous two.

There are several ways to print fibonacci series in Java language. Let’s take a detailed look at all the approaches to display fibonacci series in Java

- Fibonacci Series in Java using For Loop
- Fibonacci Series Program in Java using While Loop
- Java Program to Implement Efficient O(log n) Fibonacci Generator

In this method, we use a for loop to generate n fibonacci series.

Here is the source code of the Java Program to Generate Fibonacci Numbers. The Java program is successfully compiled and run on a Windows system. The program output is also shown below.

/* * Fibonacci Series in Java using For Loop */ import java.util.Scanner; public class Fibonacci { public static void main(String[] args) { int n, a = 0, b = 0, c = 1; Scanner s = new Scanner(System.in); System.out.print("Enter value of n:"); n = s.nextInt(); System.out.print("Fibonacci Series:"); for(int i = 1; i <= n; i++) { a = b; b = c; c = a + b; System.out.print(a+" "); } } }

1. Enter the number of terms you want as an input.

2. The main class is named **Fibonacci** and declares integer variables **n, a, b, and c**.

3. The program generates a Fibonacci series up to **n** using a for loop.

4. For each iteration of the for loop, the current value of **a** is printed and then **a, b, and c** are updated according to the formula **c = a + b**.

5. When the for loop finishes, the program prints the Fibonacci series.

In this case, entering “5” as the limit to generate the fibonacci series.

Enter value of n: 5 Fibonacci Series: 0 1 1 2 3

In this method, we use a while loop to generate n fibonacci series.

Here is the source code of the Java Program to Generate Fibonacci Numbers using while loop. The Java program is successfully compiled and run on a Windows system. The program output is also shown below.

/* * Fibonacci Series in Java using While Loop */ import java.util.Scanner; public class Fibonacci { public static void main(String[] args) { int n, a = 0, b = 0, c = 1, i = 1; Scanner s = new Scanner(System.in); System.out.print("Enter value of n:"); n = s.nextInt(); System.out.print("Fibonacci Series:"); while(i<=n) { a = b; b = c; c = a + b; System.out.print(a+" "); i++; } } }

1. The main class is named **Fibonacci** and declares integer variables **n, a, b, c, and i**.

2. The program generates a Fibonacci series up to **n** using a while loop.

3. For each iteration of the while loop, the current value of **a** is printed and then **a, b, and c** are updated according to the formula **c = a + b**.

4, The loop continues as long as **i** is less than or equal to **n**, with **i** incrementing by 1 each iteration.

5. When the loop completes, the program prints the Fibonacci series

In this case, entering “8” as the value to generate the fibonacci series.

Enter value of n: 8 Fibonacci Series: 0 1 1 2 3 5 8 13

This is a program to generate nth fibonacci number with O(log n) complexity.

Here is the source code of the Java Program to Implement Efficient O(log n) Fibonacci generator . The Java program is successfully compiled and run on a Windows system. The program output is also shown below.

/* * Java Program to Implement Efficient O(log n) Fibonacci generator */ import java.util.Scanner; import java.math.BigInteger; /** Class FibonacciGenerator **/ public class FibonacciGenerator { /** function to generate nth fibonacci number **/ public void genFib(long n) { BigInteger arr1[][] = {{BigInteger.ONE, BigInteger.ONE}, {BigInteger.ONE, BigInteger.ZERO}}; if (n == 0) System.out.println("\nFirst Fibonacci number = 0"); else { power(arr1, n - 1); System.out.println("\n"+ n +" th Fibonacci number = "+ arr1[0][0]); } } /** function raise matrix to power n recursively **/ private void power(BigInteger arr1[][], long n) { if (n == 0 || n == 1) return; BigInteger arr2[][] = {{BigInteger.ONE, BigInteger.ONE}, {BigInteger.ONE, BigInteger.ZERO}}; power(arr1, n / 2); multiply(arr1, arr1); if (n % 2 != 0) multiply(arr1, arr2); } /** function to multiply two 2 d matrices **/ private void multiply(BigInteger arr1[][], BigInteger arr2[][]) { BigInteger x = (arr1[0][0].multiply(arr2[0][0])).add(arr1[0][1].multiply(arr2[1][0])); BigInteger y = (arr1[0][0].multiply(arr2[0][1])).add(arr1[0][1].multiply(arr2[1][1])); BigInteger z = (arr1[1][0].multiply(arr2[0][0])).add(arr1[1][1].multiply(arr2[1][0])); BigInteger w = (arr1[1][0].multiply(arr2[0][1])).add(arr1[1][1].multiply(arr2[1][1])); arr1[0][0] = x; arr1[0][1] = y; arr1[1][0] = z; arr1[1][1] = w; } /** Main function **/ public static void main(String[] args) { Scanner scan = new Scanner(System.in); System.out.println("Efficient Fibonacci Generator\n"); System.out.println("Enter number n to find nth fibonacci number\n"); long n = scan.nextLong(); FibonacciGenerator fg = new FibonacciGenerator(); fg.genFib(n); } }

1. This program calculates the nth Fibonacci number using a matrix-based algorithm and the **BigInteger** class for handling large numbers.

2. The program is implemented as a class called FibonacciGenerator that has three methods: **genFib**, **power**, and **multiply**.

3. The **genFib** method uses the matrix-based algorithm to calculate the nth Fibonacci number.

4. The **power** method recursively raises a matrix to a power, while the multiply method multiplies two 2D matrices.

5. The main method prompts the user to enter the value of **n** and calls the **genFib** method to generate the **nth** Fibonacci number.

In this case, enter “1000” as the limit for generating the Fibonacci series.

Efficient Fibonacci Generator Enter number n to find nth fibonacci number 1000 1000 th Fibonacci number = 43466557686937456435688527675040625802564660517371780 40248172908953655541794905189040387984007925516929592259308032263477520968962323 9873322471161642996440906533187938298969649928516003704476137795166849228875

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