On the number of invariant straight lines for polynomial differential systems (Q1306195)

Let \(\alpha(n)\) be the maximum number of invariant straight lines possible in a polynomial differential system of degree \(n>1\) having finitely many invariant straight lines. By using a method in graph theory developed by the second author, the remarkable result: \(\alpha(5)= 14\) (together with many upper and lower bounds for \(\alpha(6),\alpha(7),\dots, \alpha(20)\)) is obtained. This result gives a negative answer to the old conjecture: \(\alpha(n)= 2n+1\), if \(n\) is even; \(\alpha(n)= 2n+2\), if \(n\) is odd. Moreover, the authors can prove that \(\alpha(n)\leq 3n-1\) for all \(n\). So, now they have the following results (the first three are known): \(\alpha(2)= 5= 3\cdot 2-1\), \(\alpha(3)= 8= 3\cdot 3-1\), \(\alpha(4)= 9< 3\cdot 4-1\), \(\alpha(5)= 14= 3\cdot 5-1\). There are interesting questions: Can we get a general formula for \(\alpha(n)\)? Is this question related to number theory?