Pascal's Triangle Generator

Pascal's Triangle is a triangular array where each number is the sum of the two numbers directly above it. Named after Blaise Pascal (1623-1662), though it was known to mathematicians in China, India, and Persia centuries earlier.

Construction Rule

Each row starts and ends with 1. Every other number is the sum of the two numbers above it:

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

Mathematical Properties

1. Binomial Coefficients

The numbers in row n are the binomial coefficients C(n,k) or "n choose k", which represent the number of ways to choose k items from n items.

C(n,k) = n! / (k! × (n-k)!)

2. Row Sum = Powers of 2

The sum of numbers in row n equals 2ⁿ:

• Row 0: 1 = 2⁰
• Row 1: 1 + 1 = 2 = 2¹
• Row 2: 1 + 2 + 1 = 4 = 2²
• Row 3: 1 + 3 + 3 + 1 = 8 = 2³

3. Fibonacci Sequence

The Fibonacci sequence can be found by summing the "shallow diagonals" of Pascal's Triangle. Starting from the top, sum along diagonal paths to get: 1, 1, 2, 3, 5, 8, 13, 21...

4. Triangle Numbers

The third diagonal contains the triangle numbers (1, 3, 6, 10, 15...), which represent the number of dots in triangular patterns.

5. Sierpiński Triangle

When you color odd and even numbers differently, a pattern similar to the Sierpiński Triangle fractal emerges. Try the "Even/Odd Numbers" highlighting to see this pattern!

Applications

• Probability Theory - Binomial probability distributions
• Combinatorics - Counting combinations and permutations
• Algebra - Binomial expansion coefficients: (a+b)ⁿ
• Number Theory - Modular arithmetic patterns
• Computer Science - Dynamic programming, algorithm analysis