Second Order Fibonacci Number in terms of Fibonacci Numbers

Theorem
The second order Fibonacci number $\mathcal F_n$ can be expressed in terms of Fibonacci numbers as:


 * $\dfrac {3 n + 3} 5 F_n - \dfrac n 5 F_{n + 1}$

Proof
Let $\mathcal G \left({z}\right) = \displaystyle \sum_{n \mathop \ge 0} \mathop F_n z^n$ be a generating function for $\mathcal F_n$.

Then we have:

Thus:

Then from Summation over k to n of Product of kth with n-kth Fibonacci Numbers, the coefficient of $z^n$ in $\left({G \left({z}\right)}\right)^2$ is:
 * $\dfrac {\left({n - 1}\right) F_n + 2n F_{n - 1} } 5$

Thus the coefficient of $z^{n + 1}$ in $z \left({G \left({z}\right)}\right)^2$ is likewise:
 * $\dfrac {\left({n - 1}\right) F_n + 2n F_{n - 1} } 5$

and so the coefficient of $z^n$ in $G \left({z}\right) + z \left({G \left({z}\right)}\right)^2$ is:


 * $F_n + \dfrac {\left({n - 2}\right) F_{n - 1} + 2 \left({n - 1}\right) F_{n - 2} } 5$

Hence: