Expected time to reach final state for a random walk on a finite grid I am not a mathematician, just a user of mathematics, so perhaps this is the wrong place to ask, but there is a problem I need some help with.
You have a $n \times n$ grid, and you start at the $(1,1)$ square. You move randomly one square on the $x$ or $y$ axis (no diagonal movements allowed). You keep moving until you reach square $(n,n)$. 
At that moment, the game ends. The variable of interest is $T$, the time it took you to reach the end position. It is clear to me that $T$ is always finite (just intuition: if you are not in $(n,n)$ you always have a chance to reach it, if you are there, you will never leave it, so if the game goes on for long enough you will end up there)
My question is about the distribution of $T$. I am interested in understanding how to define the distribution, and how to compute the expected value of $T$ as a function of $n$. Can you point me to something I could read about it? 
Thanks in advance.
 A: Let $e_n$ be the expected number of moves to get from square (1,1) to square (n,n) in an $n \times n$ grid ($n$ rows, $n$ columns), where "square $(i,j)$" refers to the $1 \times 1$ cell at row $i$, column $j$.

Assumed rule: From any given square other than square (n,n), the next square is randomly chosen from one of the squares in the grid adjacent to the given square.

I doubt that there is a closed form expression for $e_n$ in terms of $n$.

For very small $n$, it's feasible to compute $e_n$ by hand. For example, it's immediate that $e_1 = 0$, and it's easy to show that $e_2 = 4$. 

Using a Maple progam, I computed $e_n$ for $1 \le n \le 10$. Here are the results ...
\begin{align}
 e_1 &= 0 \\
 e_2 &= 4 \\
 e_3 &= 18\\
 e_4 &= 312/7 \approx 44.57142857 \\
 e_5 &= 940/11 \approx 85.45454545 \\
 e_6 &= 4684/33 \approx 141.9393939 \\
 e_7 &= 268170/1247 \approx 215.0521251 \\
 e_8 &= 110081552/360161 \approx 305.6453975 \\
 e_9 &= 31757976/76627 \approx 414.4489018 \\
 e_{10} &= 77013755100/142065451 \approx 542.1005217 \\
 \end{align}
