
COMMENTS

Without loss of generality, we may assume the central square has vertices (0,0), (1,0), (0,1), (1,1).
Suppose n >= 1. Then all nodes in the graph, whether or not in the central square, have rational coordinates with denominator at most 4*n^2 + 2*n + 1, which for n = 1, 2, 3, ... is 7, 21, 43, 73, 111, ... (cf. A054569).
This maximum is always attained, for example by the node at the intersection of the lines 2*n*x + y = n, joining (0,n) to (1, n) and x + (2*n+1)*y = n, joining (n,0) to (n+1,1).
In the central square, the maximum number of sides in any region is (for n = 0, 1, 2, 3, ...) 3, 4, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 6, 6, 6, 6, 7, 6, 7, 7, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 6, 6, ... We conjecture that 7 is the maximum.  Lars Blomberg, Sep 19 2020.
See A331456 for further illustrations.


LINKS

Lars Blomberg, Table of n, a(n) for n = 0..74
Scott R. Shannon, Colored illustration for a(0): there are 4 regions, so a(0) = 1.
Scott R. Shannon, Colored illustration for a(1): there are 56 regions, so a(1) = 14.
Scott R. Shannon, Colored illustration for a(2): there are 280 regions, so a(2) = 70.
Scott R. Shannon, Colored illustration for a(3): there are 924 regions, so a(3) = 231.
Scott R. Shannon, Black and white illustration for a(1) (Shows vertices and regions for each square)
Scott R. Shannon, Black and white illustration for a(2) (Shows vertices and regions for each square)
Scott R. Shannon, Black and white illustration for a(3) (Shows vertices and regions for each square)
Scott R. Shannon, Black and white illustration for a(4) (Shows vertices and regions for each square)
Scott R. Shannon, Black and white illustration for a(5) (Shows vertices and regions for each square)
Scott R. Shannon, Black and white illustration for a(6) (Shows vertices and regions for each square)
