Summation over k to n of Product of kth with n-kth Fibonacci Numbers

Theorem

 * $\ds \sum_{k \mathop = 0}^n F_k F_{n - k} = \dfrac {\paren {n - 1} F_n + 2n F_{n - 1} } 5$

where $F_n$ denotes the $n$th Fibonacci number.

Proof
From Generating Function for Fibonacci Numbers, a generating function for the Fibonacci numbers is:


 * $\map G z = \dfrac z {1 - z - z^2}$

Then:

where:
 * $\phi = \dfrac {1 + \sqrt 5} 2$
 * $\hat \phi = \dfrac {1 - \sqrt 5} 2$

Hence:

Thus the coefficient of $z^n$ in $\paren {\map G z}^2$ is $\ds \sum_{k \mathop = 0}^n F_k F_{n - k}$.

Hence: