Math Tools

Mathematical Principles & Theorems

Generates binomial coefficients \(\binom{n}{k}\) via Pascal's recurrence \(\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\). Binomial Theorem: \((x+y)^n = \sum_{k=0}^n \binom{n}{k}x^{n-k}y^k\). Hidden combinatorial structures: Row sums equal powers of two \(\sum \binom{n}{k} = 2^n\); Shallow diagonals sum to Fibonacci numbers \(F_{n+1} = \sum \binom{n-k}{k}\); Hockey-stick identity \(\sum_{i=r}^n \binom{i}{r} = \binom{n+1}{r+1}\); Modular arithmetic \(\binom{n}{k} \bmod p\) reveals fractal Sierpiński triangles (Lucas' Theorem).

Operating Instructions

• Set row depth \(N\) using the numeric value box.
• Select coloring mode: Value Display, Modulo \(m\) Parity Color Map, Fibonacci Diagonals, or Hockey-Stick Identity.
• Hover over individual cells to inspect index coordinates \((n, k)\), formula, and additive parents.
• Observe the emergent fractal Sierpiński gasket pattern when setting modulus \(m=2\).