Pell Number in Java

In this section, we will learn what is Pell number and also create Java programs to check if the given number is a Pell number or not. The Pell number program frequently asked in Java coding interviews and academics.

Pell Number

It is a sequence or series of numbers that starts with 0 and 1, and subsequent numbers are the sum of twice the previous Pell number and the Pell number before that. It is an OEIS sequence A000129. Sometimes it is also known as lambda numbers. It is a recursively defined sequence similar to the Fibonacci sequence. We can represent the Pell sequence mathematically, as follows:

Where n>1.

A Pell prime is a Pell number that is prime. The first few Pell primes are 2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, etc. It is an OEIS sequence A086383.

It can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combinatorial enumeration problems.

Pell Number Example

Let's see how the Pell series is generated.

We know the first and second terms of the series are 0 and 1. On the basis of these two terms, we can calculate the next Pell number.

P₀ = 0

P₁ = 1

P₂ = 1 * 2 + 0 = 2

P₃ = 2 * 2 + 1 = 5

P₄ = 5 * 2 + 2 = 12

P₅ = 12 * 2 + 5 = 29

P₆ = 29 * 2 + 12 = 70

P₇ = 70 * 2 + 29 = 169

P₈ = 169 * 2 + 70 = 408

P₉ = 408 * 2 + 169 = 985

P₁₀ = 985 * 2 + 408 = 2378

We have calculated the first ten terms of the Pell series. Similarly, we can calculate other terms such as 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428, 38613965, 93222358, 225058681, 543339720, 1311738121, 3166815962, 7645370045, 18457556052, 44560482149, 107578520350, 259717522849.

Steps to Find Pell number

Initialize the first two terms of the series as P₀=0 and P₁=1.
In order to calculate the third (P₂) term, do the following:
Multiply the previous term (P₁) by 2 and add the Pell number before that i.e. P₀. Hence, we get the third term of the series.
Repeat step 2 up to the number of terms we want to calculate.

Let's implement the above steps in a Java program.

Pell Number Java Program

Using Recursion Approach

PellNumberExample1.java

public class PellNumberExample1
{
//function to find the nth pell number
public static int pellNumber(int num)
{
if (num <= 2)
return num;
//recursively called the function
return 2 * pellNumber(num - 1) + pellNumber(num - 2);
}
//driver code
public static void main(String args[])
{
//num is a term that we want to find
int num = 4;
//calling the function and prints the pell number 
System.out.println("The " +num+"-th" +" term of the series is: " +pellNumber(num));
}
}

Output:

The 7-th term of the series is: 169

Let's see another approach to find the Pell numbers using an iterative approach.

Using Iterative Approach

PellNumberExample2.java

public class PellNumberExample2
{
//function finds the nth Pell number
public static int pellNumber(int num)
{
if (num <= 2)
return num;
//a and b denote first (P1) and second (P2) term
int x = 1;
int y = 2;
int z;
for (int i = 3; i <= num; i++) 
{
z = 2 * y + x;
x = y;
y = z;
}
return y;
}
//driver code
public static void main(String args[])
{
int num = 17;
System.out.println("The " +num+"-th" +" term of the series is: " +pellNumber(num));
}
}

Output:

The 17-th term of the series is: 1136689

Print All Pell Numbers up to Specified Limit

PellNumberExample3.java

import java.util.Scanner;
public class PellNumberExample
{
public static void main(String args[])
{
int n, limit, a=1, b=0, c;
Scanner sc=new Scanner(System.in);
System.out.print("Enter the limit: ");
limit=sc.nextInt();
System.out.println("First "+limit+" Pell numbers are: ");
for(n=1; n<=limit; n++)
{
c= a + 2 * b;
System.out.print(c+" ");
a = b;
b = c;
}
}
}

Output:

Enter the limit: 20
First 20 Pell numbers are: 
1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428

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