I need to prove that any integer can created using a sum of elements in $a_{n}=2a_{n-1}-\left\lfloor \frac{a_{n-1}}{2}\right\rfloor ,a_{1}=1$ without repetition.
My attempt was just bad...
I've tried proving by induction but it I'm not sure it's legitimate.
I've divided the proof into 2 cases:
- $k=1$ then it's obvious since $a_1 =1 $ -$k\geq2$ by induction let's prove that any $k\geq2\in\mathbb{N}$ can be created using the above mentioned series without using $a_1$. For $k=2$ it works since $a_2=2$ and given it's correct for some $k$, for $k+1$ we use the I.H for $k$ and then use $a_1=1$ since we did not use it. It's bad...
I've also tried looking at $a_{n+1}-a_n$ but it didn't help.
Two main questions:
- Any clues to how would I prove it?
- I've tried to understand how to find a general non-recursive formula for this series and couldn't( since it's not linear and not anything I've learned so far...). This part is actually pure curiosity and is not homework but it really is more interesting :)