Factorials

This is the sequence of factorials. Each factorial is made by putting it and all the numbers before it into one big tower. Here is the number line from 1 to 9.

By looking at the colours one at a time, you can work out how many blocks of each colour there is in a factorial. Here are only the blue blocks in the number line from 1 to 20

The number of blue blocks in n factorial is

$\lfloor \frac{n!}{2} \rfloor + \lfloor \frac{n!}{4} \rfloor + \lfloor \frac{n!}{8} \rfloor + \ldots$