# Finding an Explicit Formula from the Recurrence: $na_{n}= 2 ( a_{n-1}+a_{n-2})$

Here is the recurrence:

$$na_{n}=2(a_{n-1}+a_{n-2}) \qquad\text{ where } a_{0}=1\text{ and }a_{1}=2$$

At first I thought that this could be easily solved by simply multiplying the Fibonacci generating sequence by $\frac{2}{n}$, however I quickly discovered it was not this simple. I calculated some values and saw the following:

\begin{align*} a_{0}&=1\\ a_{1}&=2\\ a_{2}&=3\\ a_{3}&=\frac{10}{3}\\ a_{4}&=\frac{19}{6}\\ a_{5}&=\frac{13}{5}\\ a_{6}&=\frac{173}{90} \end{align*}

I cannot (for the life of me!) figure out a pattern amongst these numbers. I was pretty confident about those values, but I could have made an arithmetic mistake that would account for my not being able to find a pattern...?? Any and all help is greatly appreciated. Thanks!

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