0s and 1s
In Pascal’s triangle, we see that the inner numbers form a pattern depending
on whether they are even or odd. If we substitute 1 for the odd numbers and 0
for the even numbers we get a representation which is the same pattern as the
remarkable fractal known as the Sierpinski gasket (see page 102).
Adding signs
We can write down the Pascal triangle that corresponds to the powers of (−1
- x), namely (−1 + x)n.
Adding signs
In this case the triangle is not completely symmetric about the vertical line,
and instead of the rows adding to powers of 2, they add up to zero. However it is
the diagonals which are interesting here. The southwestern diagonal 1, −1, 1,
−1, 1, −1, 1, −1,... are the coefficients of the expansion while the terms in the
next diagonal along are the coefficients of the expansion
(1 + x)−1 = 1 − x + x^2 − x^3 + x^4 − x^5 + x^6 − x^7 +...