Points at distance $k$ or less in 2D grid How can we prove that the total number of mesh points k or fewer jumps away from an arbitrary point in a 2D mesh is: $2k^2 + 2k + 1$.

I tried making a $3 \times 3$ mesh. 

     a --- b --- c
     |     |     |
     d --- e --- f
     |     |     |
     g --- h --- i


For point $e$, the number of points $k=1$ jump away is four: $b$, $d$, $f$, $h$.
But, according to the formula, it should be $2\times1^2 + 2\times1 + 1=5$.
I am stuck!
Read your question more carefully. The formula you give is for


  the total number of mesh points $k$ or fewer jumps away


(my emphasis).  And there are indeed five points one jump or fewer away from&nbsp;$e$: $b$, $d$, $f$, $h$ and $e$&nbsp;itself.