Sylvester Sequence in Java

In this section, we will learn what is Sylvester Sequence and also create Java programs to that calculates the Sylvester Sequence. The Sylvester Sequence program is frequently asked in Java coding interviews and academics.

Sylvester Sequence

A Sylvester Sequence is a number sequence where each term is the product of the previous terms, plus 1. The first two terms of the sequence are 2 and 3. Using the first two terms, we will compute the other terms of the Sylvester sequence.

S₁ = 2, and S₂ = 3

Thus,

S₃ = (S₁ x S₂) + 1 = (2 x 3) + 1 = 7

S₄ = (S₁x S₂ x S₃) + 1 = (2 x 3 x 7) + 1 = 43

S₅ = (S₁x S₂ x S₃ x S₄) + 1 = (2 x 3 x 7 x 43) + 1 = 1807

S₆ = (S₁x S₂ x S₃ x S₄ x S₅) + 1 = (2 x 3 x 7 x 43 x 1807) + 1 = 3263443

Let's see the different approaches for computing the Sylvester Sequence.

Approach: Using Nested Loop

Observe the following program for finding the Sylvester sequence.

FileName: SylvesterSeq.java

// important import statement
import java.util.ArrayList;

public class SylvesterSeq 
{
// main method
public static void main(String argvs[]) 
{
// creating a list for storing the Sylvester sequence
ArrayList<Integer> al = new ArrayList<Integer>();

// the first two terms
al.add(2);
al.add(3);

int n = 6; // for computing the first six terms

System.out.println("The first " + n + " terms of the Sylvester's Sequence are: ");

for(int i = 0; i < n; i++)
{
int ans = 1;

if((i == 0) || (i == 1))
{
// for printing the first two terms of the sequence
System.out.print(al.get(i) + " ");
continue;
}

// computing the other terms (starting from the 3rd Term)
for(int j = 0; j < al.size(); j++)
{
ans = ans * al.get(j);
}

ans =  ans + 1;
al.add(ans);

// printing the other terms of the sequence
System.out.print(ans + " ");
}


}
}

Output:

The first 6 terms of the Sylvester's Sequence are: 
2 3 7 43 1807 3263443

Complexity Analysis: Since the program is using the nested loop, the time complexity of the program is O(n2). Also, the program uses an array list, making the space complexity of the program O(n).

Let's see the optimized way of computing the Sylvester Sequence, where we will be reducing the time as well as the space complexity.

FileName: SylvesterSeq1.java

// important import statment
import java.util.ArrayList;

public class SylvesterSeq1 
{

public void displaySequence(int l)
{
int a = 1; // for storing the product.
int answer = 2; // for storing the curr number.

// Loop from 1 to l.
for (int i = 1; i <= l; i++) 
{
System.out.print(answer + " ");
answer = a * answer;
a = answer;
answer = (answer + 1);
}
}

// main method
public static void main(String argvs[]) 
{
// creating an object of the class SylvesterSeq1
SylvesterSeq1 obj = new SylvesterSeq1();

int n = 6; // for computing the first six terms

System.out.println("The first " + n + " terms of the Sylvester's Sequence are: ");

obj.displaySequence(n);


}
}

Output:

The first 6 terms of the Sylvester's Sequence are: 
2 3 7 43 1807 3263443

Complexity Analysis: The time complexity of the program is O(n), and the space complexity of the program is O(1).

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Sylvester Sequence in Java