$n$ balls numbered from $1$ to $n$ 
If we consider a box containing $n$ balls numbered from $1$ to $n$. If $455$
  is the number of ways to get three balls from the box such that no two
  balls are consecutively numbered, then we have to find the value of $n$.

Someone please help me out in this. I am not getting anything, how to start it?

As I tend to make mistakes with this kind of problems, I asked my friend Ruby to do some test runs. She is pretty fast at counting and makes no silly mistakes if I explained it well.
We got different numbers, much larger numbers until I realized that order should not matter for a draw, so drawing $1,3,5$ is considered the same as drawing $5,3,1$. Then Ruby found $455$ non-consecutive draws for $n=17$ balls, confirming Arthur's formula.
Here is what I told her: balls.rb
This is what she found: balls.log
