Pascal Triangle In Java Using While Loop

Generating Pascal's Triangle is a classic programming exercise that demonstrates pattern recognition and iterative problem-solving. It's an array of binomial coefficients, where each number is the sum of the two numbers directly above it.

In this article, you will learn how to generate Pascal's Triangle in Java using while loops, providing a step-by-step implementation.

Problem Statement

Pascal's Triangle is a triangular array of the binomial coefficients. It begins with a single '1' at the top, and each subsequent number is the sum of the two numbers directly above it. If there is only one number above, it's considered 0. The first few rows look like this:

Row 0: 1 Row 1: 1 1 Row 2: 1 2 1 Row 3: 1 3 3 1 Row 4: 1 4 6 4 1

The problem is to write a Java program that takes an integer numRows as input and prints the first numRows of Pascal's Triangle using while loops.

Example

For numRows = 5, the expected output of Pascal's Triangle would be:

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1

Background & Knowledge Prerequisites

To understand and implement this solution, you should have a basic understanding of:

• Java Basics: Variables, data types, while loops, conditional statements (if-else).
• Lists in Java: How to declare, add elements to, and access elements from ArrayList.
• Input/Output: Basic console input and output using System.out.println().

For this specific implementation, we will utilize java.util.ArrayList to store the rows of the triangle.

Use Cases or Case Studies

Pascal's Triangle appears in various fields beyond just mathematical curiosities:

• Combinatorics and Probability: The numbers in Pascal's Triangle represent binomial coefficients, which are crucial for calculating combinations (e.g., "n choose k") and probabilities.
• Polynomial Expansion: The numbers in each row are the coefficients in the expansion of binomials like (x + y)^n.
• Computer Science Algorithms: Its patterns are sometimes used in algorithms related to dynamic programming or graph theory.
• Fractals: Connecting the odd numbers in Pascal's Triangle reveals patterns similar to the Sierpinski Triangle.

Solution Approaches

We will focus on an iterative approach using while loops to build the triangle row by row.

Generating Pascal's Triangle with a While Loop

This approach iteratively builds each row of the triangle based on the values of the previous row, utilizing nested while loops.

// Pascal's Triangle using While Loop
import java.util.ArrayList;
import java.util.List;
import java.util.Scanner;

// Main class containing the entry point of the program
public class Main {
    public static void main(String[] args) {
        // Step 1: Initialize a Scanner for user input
        Scanner scanner = new Scanner(System.in);
        System.out.print("Enter the number of rows for Pascal's Triangle: ");
        int numRows = scanner.nextInt();
        scanner.close();

        // Step 2: Create a list to store all rows of the triangle
        List<List<Integer>> pascalTriangle = new ArrayList<>();

        // Step 3: Outer while loop to iterate through each row
        int currentRowNum = 0;
        while (currentRowNum < numRows) {
            // Create a list for the current row
            List<Integer> currentRow = new ArrayList<>();
            
            // Inner while loop to populate elements of the current row
            int elementIndex = 0;
            while (elementIndex <= currentRowNum) {
                // The first and last elements of each row are always 1
                if (elementIndex == 0 || elementIndex == currentRowNum) {
                    currentRow.add(1);
                } else {
                    // For other elements, sum the two numbers from the previous row
                    // specifically, the element at (elementIndex-1) and (elementIndex)
                    List<Integer> prevRow = pascalTriangle.get(currentRowNum - 1);
                    int sum = prevRow.get(elementIndex - 1) + prevRow.get(elementIndex);
                    currentRow.add(sum);
                }
                elementIndex++;
            }
            
            // Add the completed current row to the pascalTriangle list
            pascalTriangle.add(currentRow);
            
            currentRowNum++; // Move to the next row
        }

        // Step 4: Print the Pascal's Triangle
        int i = 0;
        while (i < pascalTriangle.size()) {
            List<Integer> rowToPrint = pascalTriangle.get(i);
            int j = 0;
            while (j < rowToPrint.size()) {
                System.out.print(rowToPrint.get(j) + " ");
                j++;
            }
            System.out.println(); // Move to the next line after printing a row
            i++;
        }
    }
}

Run

Sample Output

If the user enters 5 for the number of rows:

Enter the number of rows for Pascal's Triangle: 5
1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1

Stepwise Explanation

1. User Input: A Scanner object is used to prompt the user for numRows, which determines how many rows of the triangle to generate.
2. Triangle Storage: An ArrayList of ArrayList called pascalTriangle is created to store all generated rows. Each inner ArrayList represents a single row of the triangle.
3. Outer while Loop (Rows):

• This loop runs numRows times, with currentRowNum ranging from 0 to numRows - 1, representing the current row being constructed.
• Inside this loop, a new ArrayList named currentRow is initialized for each row.

1. Inner while Loop (Elements):

• This loop populates the currentRow. elementIndex ranges from 0 up to currentRowNum.
• Base Cases: If elementIndex is 0 (the first element of any row) or elementIndex is currentRowNum (the last element of the current row), the value is always 1.
• Calculating Other Elements: For any other element, its value is the sum of two elements from the previous row. These elements are located at elementIndex - 1 and elementIndex in the pascalTriangle.get(currentRowNum - 1) (which retrieves the previous row).
• Each calculated value is added to currentRow.

1. Adding and Incrementing: After the inner loop completes, the currentRow is fully populated and added to the pascalTriangle. Then currentRowNum is incremented to move to the next row.
2. Printing the Triangle: A final set of nested while loops iterates through pascalTriangle to print each row and its elements, ensuring each row is printed on a new line.

Conclusion

Generating Pascal's Triangle using while loops in Java is an excellent way to practice iterative logic and list manipulation. The core idea relies on the simple property that each number is the sum of the two numbers above it, with edge cases handled for the '1's at the beginning and end of each row.

Summary

• Pascal's Triangle is a triangular array where numbers are the sum of the two above.
• The while loop is effective for iterative row-by-row construction.
• java.util.ArrayList is suitable for storing rows and elements.
• Each row starts and ends with '1'.
• Intermediate elements are calculated by summing corresponding elements from the previous row.
• Nested while loops handle row iteration and element calculation within each row.