Pascal's Triangle and Its Relatives

Exercises

Pascal's triangles for other polynomials

1. Pascal's triangles for (1 + x + x²)ⁿ. We shade the boxes containing numbers congruent to

1 (mod 2)	1 or 2 (mod 3)

First observe these graphs grow at twice the horizontal rate as the standard Pascal's triangle. This is no surprise, because the polynomial for each row is obtained from the one above by multiplying by (1 + x + x²). The x² term guarantees each row contains two more boxes than the row above it. Consequently these shapes cannot be identical with the standardPascal's triangles: at the very least, they will be distorted versions of the standard.

The left picture is not a distortion of the standard Pascal's triangle shading the odd terms, but the right picture is a distortion of the standard Pascal's triangle shaading the terms congruent to 1 or 2 (mod 3).

Return to Other polynomials Exercises.