Finding the function of these numbers $1, 2, 5, 13, 34, 89, 233, 610$ Firstly I used the differences between them but I found the numbers return again.
How can I find the function of these numbers
 A: Too long for a comment; the method for detecting polynomial sequences is the calculus of differences and well documented. Conway and Guy give (not their invention) a variant that detects sequences based on exponentials, including Fibonacci or every other Fibonacci as you have here. I know i photocopied a few pages giving the technique, I will see if I can find them. Note that the book is extremely informal. 
A: $a_0=1,a_1=2$ and for $n\geq2$ $a_n=3a_{n-1}-a_{n-2}$
A: I happened to notice each value is twice the previous value plus all the values before that. For example, $$13=2(5)+2+1$$ The sequence has $a_1=1$ and $$a_n=2a_{n-1}+\sum_{k=1}^{n-2}a_k$$
But then $$a_{n-1}=2a_{n-2}+\sum_{k=1}^{n-3}a_k$$ and subtracting left sides and right sides: $$a_n-a_{n-1}=2a_{n-1}-a_{n-2}$$ So $$a_n=3a_{n-1}-a_{n-2}$$ and $$a_{n-1}=a_{n-1}$$ So $$
\begin{align}\begin{bmatrix}a_n\\ a_{n-1}\end{bmatrix}&=\begin{bmatrix}3&-1\\1&0\end{bmatrix}\begin{bmatrix}a_{n-1}\\ a_{n-2}\end{bmatrix}\\
&=\begin{bmatrix}3&-1\\1&0\end{bmatrix}^{n-2}\begin{bmatrix}a_{2}\\ a_{1}\end{bmatrix}\\
&=\begin{bmatrix}3&-1\\1&0\end{bmatrix}^{n-2}\begin{bmatrix}2\\ 1\end{bmatrix}\\
\implies a_n&=\begin{bmatrix}1&0\end{bmatrix}\begin{bmatrix}3&-1\\1&0\end{bmatrix}^{n-2}\begin{bmatrix}2\\ 1\end{bmatrix}\\
\end{align}$$
If you want, you can diagonalize the matrix and express the formula for $a_n$ as a linear combination of powers of the eigenvalues.
A: Let $n_{1} = 1$. Then,
$n_{k+1} = n_{k}+ \sum_{j=1}^{k} n_{j}, $ for $k \geq 1$.
A: 1053253    
${{
\left({1+\sqrt5}\over2\right)^n
-\left({1-\sqrt5}\over2\right)^n
}\over\sqrt5}$, where $n$ is of odd parity
A: this is the positive numbers of the Fibonacci numbers in Z.
What I mean:
public class MySeedFibbo {
public static void main(String[] args) {
    List<Integer> init = Arrays.asList(1, 1);
    Stream<List<Integer>> negativeFib = Stream.iterate(init, l -> Arrays.asList(l.get(1) - l.get(0),l.get(0)));
    negativeFib.limit(15).forEach(l->System.out.println(l.get(1)));
    }
}

Result if you run this code in Java SE 8 :
0
1
-1
2
-3
5
-8
13
-21
34
-55
89
-144
233
-377
610
-987
1597
Modify the last line like this:
negativeFib.limit(20).forEach(l->{ if(l.get(1)>0) System.out.println(l.get(1));});

And we get only the positive numbers which form the sequence you're looking for. 
