Represent the set $R_{n\times n}=\{1,2,\ldots, n\}\times\{1,2,\ldots, n\} $ as a rectangle of $n$ by $n$ points as in the figures below for example. How to calculate the number of circuits that visit a chosen point in this rectangle? What is the most visited point on this rectangle?

Making the question more precise we fix the settings below.

A circuitc $c$ in rectangle $R_{n\times n}$ is defined as a sequence of distinct points $x_{i_1j_1}, x_{i_2j_2},x_{i_2j_2},\ldots, x_{i_{m-1}j_{m-1}}$ and a point $x_{i_{m}j_{m}}=x_{i_{1}j_{1}}$ in $R_{n\times n}$ such that $|i_{k}-i_{k+1}|+|j_{k}-j_{k+1}|=1$ for $k=1,\ldots, m-1$.

A circuit $c=\{x_{i_1j_1}, x_{i_2j_2},x_{i_2j_2},\ldots, x_{i_{m-1}j_{m-1}},x_{i_{m}j_{m}}\}$ visit a point $x_{ij}$ if $x_{ij}\in c$.

Let $N_n(x_{ij})$ the total circuits in $R_{n\times n}$ who visit the point $x_{ij}$ .

Question.How to calculate the value of $N_n(x_{ij})$? If we draw a circuit randomly which point $x_{ij}$ has the highest probability to be visited?

Calculations case $n = 1,2,3$ are trivial. I'm trying to calculate the number $N_n(x_{ij})$ by a recursive procedure which reduces the calculation for the case $n$ to $n-1$. But I think the use of this recursion depends on some well-crafted trick that escaped my attempts.