Tagged Questions

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How do I proceed from here on finding the Binet's formula via generating functions?

So, I'm stuck with the algebra for the nth number on the Fibonacci sequence in here. I managed to get to the part where $G(x) = \frac{x}{1-x-x^2}$ $=$ $\frac{x}{(1-\alpha x)(1-\beta x)}$, and I know ...
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Find a sequence

Find the function for the sequence $a_0 = 0, a_1 = 1$ and $a_{n}=a_{n+10}+a_n$ for all $n>0$.
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Closed form for the sum of even fibonacci numbers?

I recently took a look a project euler, and I am trying to think of a smart way to do number 2. I looked at the sequence, and I saw that the question is basically asking for $$\sum_{i=1}^n F_{3i}$$ ...
Could someone please tell me the closed form solution of the equation below. $$F(n) = 2F(n-1) + 2F(n-2)$$ $$F(1) = 1$$ $$F(2) = 3$$ Is there any way it can be easily deduced if the closed form ...
The nth term of the Fibonacci series is given by $F_{n}$=$\Big\lfloor\frac{\phi^{n}}{\sqrt{5}}+\frac{1}{2}\Big\rfloor$ How do you get the following expression for n from this? ...