There are many summations involving Fibonacci numbers which Mathematica 10.4 is able to evaluate directly in terms of Fibonacci numbers. For example, Mathematica evaluates the summation given below as $F_{2 (n+1)}$.
Sum[Fibonacci[2 i + 1], {i, 0, n}]
However, there are many summations involving Fibonacci numbers which Mathematica 10.4 can evaluate in closed-form, but which Mathematica apparently cannot directly evaluate in terms of Fibonacci numbers. For example, consider the following summation:
Sum[Fibonacci[i]*(-1)^i, {i, 0, n}]
Mathematica 10.4 provides the following evaluation of this expression:
$$\frac{(-2)^{-n} \left(1+\sqrt{5}\right)^{-n-1} \left(2 \left(1+\sqrt{5}\right)^{2 n}+\left(3+\sqrt{5}\right) (-4)^n-\left(5+\sqrt{5}\right) \left(-2-2 \sqrt{5}\right)^n\right)}{\sqrt{5}}$$
However, there is a very simple evaluation of the above alternating sum in terms of the Fibonacci sequence:
$$\sum _{i=0}^n (-1)^i F_i =(-1)^n F_{n-1}-1$$
Similarly, Mathematica 10.4 is not able to evaluate summations such as $\sum _{i=0}^n F_{3 i}$ in terms of Fibonacci numbers.
So it is natural to ask: are there any known algorithms or Mathematica packages for converting closed-form evaluations of sums involving Fibonacci numbers in terms of Fibonacci/Lucas numbers?
Simplify[FunctionExpand[Fibonacci[n]], n ∈ Integers]
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