cp's OEIS Frontend

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.