Successively longer sums of consecutive Fibonacci numbers: pattern?

Question

Consider the following: $$\begin{align} F_{n-1}+F_{n-2}&=F_n\\ F_{n-1}+F_{n-2}+F_{n-3}&=F_{n-1}+F_{n-1}\\ &=2F_{n-1}\\ F_{n-1}+F_{n-2}+F_{n-3}+F_{n-4}&=F_n+F_{n-2}\\ &=L_{n-1}\\ F_{n-1}+F_{n-2}+F_{n-3}+F_{n-4}+F_{n-5}&=F_{n-1}+L_{n-2}\\ &=F_{n-1}+F_{n-1}+F_{n-3}\\ &=2F_{n-1}+F_{n-3}\\ F_{n-1}+F_{n-2}+F_{n-3}+F_{n-4}+F_{n-5}+F_{n-6}&=2(F_{n-1}+F_{n-4}) \end{align}$$ Is there some logic to this pattern? Can it be predicted? Is there a formula that can be used to compute the value of $$\sum_{i=1}^{r}{F_{n-i}}$$ in terms of only Fibonacci and Lucas numbers or their multiples, no constant terms, without resorting to a summation?

Answer 1

Using the formula that I found in this question, I have derived the following: $$\begin{align} f(n,r)&=\sum_{i=1}^r{F_{n-i}}\\ &=\sum_{i=1}^{n-1}{F_i}-\sum_{i=1}^{n-r-1}{F_i}\\ &=F_{n-1+2}-1-F_{n-r-1+2}+1\\ &=F_{n+1}-F_{n-r+1}\\ \\ f(8,6)&=F_{8+1}-F_{8-6+1}\\ &=F_9-F_3\\ &=34-2\\ &=32\\ &=13+8+5+3+2+1\\ &=F_7+F_6+F_5+F_4+F_3+F_2\\ \\ f(7,3)&=F_8-F_5\\ &=21-5\\ &=16\\ &=8+5+3\\ &=F_6+F_5+F_4 \end{align}$$

So it turns out the solution is the difference of two Fibonacci numbers.

Answer 2

$$F_n = \left(\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^n\right)_{2,1}$$

which let's us treat it like any other summation except using matrices instead of scalar numbers, just remember that matrix multiplication doesn't commute and everything else is the same.

Using $X = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, proceed:

$$\begin{align} S &= \sum_{i=1}^r F_{n - i} \\ &= \left(\sum_{i=1}^r X^{n-i}\right)_{2,1}\\ &= \left(\sum_{i=n - r}^{n - 1} X^i\right)_{2,1} \\ &= \left(X^{n - r} \underbrace{\sum_{i=0}^{r - 1} X^i} _\text{Q = Geometric Series} \right)_{2,1}\\ \end{align}$$

Geometric series,

$$\begin{align} Q &= \sum_{i=0}^{r - 1} X^i\\ &= \left(X^{r} - I\right) \left(X - I\right)^{-1}\\ &= \left(X^{r} - I\right) X\\ &= X^{r+1} - X \end{align}$$

$$\begin{align} S &= \left(X^{n - r}\right) \left(X^{r+1} - X\right)_{2,1}\\ &= \left(X^{n+1} - X^{n+1-r}\right)_{2,1}\\ &= F_{n + 1} - F_{n + 1 - r} \end{align}$$