## Fibonacci Formulae

Whilst doodling with the Fibonacci sequence

 n Fn 0 1 2 3 4 5 6 7 8 9 10 … 0 1 1 2 3 5 8 13 21 34 55 …

I found some interesting formulae:

• $F_n = \displaystyle\sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-1-k}{k}$
• $F_{2m} = - \dfrac{1}{2} \displaystyle\sum_{k=0}^{2m-1} (-1)^k \binom{2m}{k} F_k$
• $\displaystyle\sum_{k=0}^{2m} (-1)^k \binom{2m+1}{k} F_k = 0$

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.

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### One Response to “Fibonacci Formulae”

1. James Malvi Says:

This tool may help mathematician to convert few numbers.
http://codebeautify.org/all-number-converter