Whilst doodling with the Fibonacci sequence

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | … |
---|---|---|---|---|---|---|---|---|---|---|---|---|

F_{n} |
0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | … |

I found some interesting formulae:

The first of these is not new, but I did not find the other two on the web.

There’s more information about the Fibonacci sequence and the binomial coefficients on Wikipedia.