Pascal's triangle: each entry is the sum of the two entries above it. Row n contains binomial coefficients C(n,0), C(n,1), ..., C(n,n). Row sums are powers of 2. The triangle encodes combinations, binomial expansion, Fibonacci numbers (diagonal sums), and fractal patterns (Sierpinski triangle from odd/even).
How to Use
Set the number of rows:
- Rows: How many rows to display
- Highlight: Even/odd pattern
- Lookup: Specific C(n,k)
Hidden Patterns
- Row sum = 2ⁿ
- Diagonal 1: natural numbers
- Diagonal 2: triangular numbers
- Fibonacci: sum of shallow diagonals
Binomial Theorem
(a+b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ. Row n of Pascal's triangle gives the coefficients. Example: (a+b)³ = 1a³ + 3a²b + 3ab² + 1b³.
Step-by-Step Instructions
- 1Set the number of rows.
- 2View the triangle.
- 3Toggle pattern highlighting.
- 4Look up specific C(n,k).
- 5Check row sums.