Is there a pattern of the length between one even Fibonacci number and another?

Question

I had seen a math problem asking for the sum of all even Fibonacci numbers up to 4 million, but I still need to know this: Is there an obvious pattern of the distance between a even Fibonacci number and the next?

Answer 1

You may notice for first that $F_n\equiv 0\pmod{2}\Longleftrightarrow n\equiv 0\pmod{3}$, then: $$ 2\sum_{k=1}^{K}F_{3k} = \sum_{k=1}^{K}\left(F_{3k-2}+F_{3k-1}+F_{3k}\right)=\sum_{n=1}^{3K}F_n = F_{3K+2}-1 \tag{1}$$ and: $$\sum_{k=1}^{K}F_{3k}=\frac{F_{3K+2}-1}{2}.\tag{2} $$ Since the last Fibonacci number less than four millions is $F_{33}=3524578$, the answer (i.e. the sum of every non-negative and even Fibonacci number less than $4\cdot 10^6$) is given by: $$\frac{F_{35}-1}{2}=\color{red}{4613732}.\tag{3}$$