The average number of regions N(n) into which n lines divide a square is N^_(n) = 1/16 n(n - 1) π + n + 1 (Santaló 1976; Finch 2003, p. 481). The maximum number of sequences is presumably the same as for circle division by lines, namely N_max(n) = 1/2(n^2 + n + 2). For n = 1, 2, ..., this gives 2, 4, 7, 11, 16, 22, ... (OEIS A000124), which is the same number into which the plane, a circle, etc. can be divided.
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