Fibonacci Formulae

Whilst doodling with the Fibonacci sequence

n 0 1 2 3 4 5 6 7 8 9 10
Fn 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.



One Response to “Fibonacci Formulae”

  1. James Malvi Says:

    This tool may help mathematician to convert few numbers.

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