Im trying to find out a general term for the following summation of products of fibonacci numbers:--
$\sum_{k=4}^{n+1} F_{k}F_{n+5-k}\; , n \geq 3$
I tried using Binet's equation but I am getting stuck at a certain point. So, I would be very glad if someone could post an answer to my question with a detailed explanation.
Here are the first few values of the summation for different values of n :--
n = 3 , ans = 9
n = 4 , ans = 30
n = 5 , ans = 73
n = 6 , ans = 158
Note : I have used the usual fibonacci notation. i.e
$ F_0=0,\;F_1=1,\;F_2=1,\;F_3=2,\;...etc $
EDIT
After reading the comments for this question I tried solving it to form a recurrence relation and this is what i ended up with :--
$ \begin{align*} G(n)&=\sum_{k=4}^{n+1} F_kF_{n+5-k}\; , n \geq 3\\ G(n)-G(n-1)&=\sum_{k=4}^{n+1} F_kF_{n+5-k}-\sum_{k=4}^{n} F_kF_{n+4-k}\\ &=F_{n+1}F_{4}+\sum_{k=4}^{n}F_kF_{n+3-k}\\ &=F_{n+1}F_{4}+F_{n}F_{3}+\sum_{k=4}^{n-1}F_kF_{n+3-k}\\ \\ &=F_{n+1}F_{4}+F_{n}F_{3}+G(n-2)\\ \\ G(n)&=G(n-1)+G(n-2)+F_{n+1}F_{4}+F_{n}F_{3}\\ \\ \end{align*} $
Is this correct ? And how do I reduce it further ?