Pascal's Triangle and Its Relatives

Exercises

Pascal's triangles of some other polynomials

1. Generate Pascal's triangle for (1 + x + x2)n. Shade the boxes containing odd numbers. Is this the standard right isosceles Sierpinski gasket?

Now shade the boxes containing numbers congruent to 1 or 2 (mod 3). Compare this with the corresponding picture for the standard Pascal's triangle.

Answer

2. Generate Pascal's triangle for (1 + x + x2)(1 + x)n. Shade the boxes containing odd numbers. Compare this with the corresponding picture for the standard Pascal's triangle.

Now shade the boxes containing numbers congruent to 1 or 2 (mod 3). Compare this with the corresponding picture for the standard Pascal's triangle.

Answer

3. Generate Pascal's triangle for xn(1 + x)n. Shade the boxes containing odd numbers. Compare this with the corresponding picture for the standard Pascal's triangle.

Now shade the boxes containing numbers congruent to 1 or 2 (mod 3). Compare this with the corresponding picture for the standard Pascal's triangle.

Answer

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