Partition of the plane by lines

It’s an exercise in induction to prove that lines in general position divide the plane into latex regions, and this number of regions is the maximal possible. Here is a partition that realizes :

Three lines can also divide the plane into 6 regions instead of 7: this happens if the triangle collapses to a point, or if two of the lines are made parallel. However, 3 lines can never divide the plane into 5 regions.

Define to be the smallest integer such that for any integer there is a partition of the plane into regions by lines. So, , and of course and . Here are a few more values of , with examples in lieu of proofs.

The last one, unattainability of 23 with 8 lines, isn’t easy to prove.

Does have a closed form? 2,3,6,8,12,15,18,24 is not in OEIS, unlike the upper bound.

Update: the 1993 paper Classification of arrangements by the number of their cells by Nicola Martinov gives a complete description of the pairs for which there is a partition of (projective) plane by lines into regions. In the affine version considered above we should simply says that the ideal line is also a part of arrangement; thus, 3-line affine arrangements correspond to 4-line projective arrangements, etc. However, it is not entirely trivial to get out of Martinov’s formulas.