Resistor matrices (square matrix)

The three matrices of resistors have inspired me to consider the $n\times n$ case. So far, I have reduced the problem to quite a neat set of recursive relations. $$ V = 2(I_1 + I_2 + \dots + I_n),\; I = 2I_1 $$ We then have $I_k = \beta_kI_{k-1}$, with $\beta_n =\frac{1}{2}$ and $\beta_{k-1} = \frac{2}{4 - \beta_k}$ These together give $$ R = 1 + \beta_2 + (\beta_2\beta_3) + \dots +(\beta_2\beta_3\beta_4\dots\beta_n) $$

So for example, in the $2\times 2$ case, we get $\beta_2 = \frac{1}{2}$, and $\beta_i =0\; \forall\;i\gt 2$. This gives $$R = 1 + \frac{1}{2}= \frac{3}{2}\Omega$$

In the $3\times 3$ we get $\beta_3 = \frac{1}{2}, \beta_2 = \frac{2}{4-\frac{1}{2}}, \beta_i =0\; \forall\;i\gt 3$, giving $$ R_{3\times 3} = 1 + \beta_2 + \beta_2\beta_3 = 1 + \frac{4}{7} + \frac{2}{7} = \frac{13}{7}\Omega $$

Derivation of these equations to follow!