%I
%S 1,2,3,6,8,12,15,18,24,28,32,36,45,50,55,60,65,78,84,90,96,102,108,
%T 126,133,140,147,154,161,168,192,200,208,216,224,232,240,248,279,288,
%U 297,306,315,324,333,342,351,390,400,410,420,430,440,450,460,470,480,528,539,550,561,572,583,594,605,616,627,638,696,708,720
%N Ivanov's number a(n) = i*(n+2i) where i = min{j > 0  j*(j+1) >= 2*(n1)}.
%C The maximum number of regions into which n lines can divide the plane is A000124(n) = n(n+1)/2 + 1.
%C Let m(n) be the least number such that every integer in the interval [m(n),n(n+1)/2 + 1] occurs as the number of regions into which n lines can divide the plane. Ivanov (2010, Theorem, p. 888) proved the upper bound m(n) <= a(n).
%C Ivanov's upper bound is sharp, i.e., m(n) = a(n), at least for n <= 6. For example, the numbers of regions into which some configuration of 6 lines divides the plane are 7, 12, 15, 16, 17, 18, 19, 20, 21, 22, 22 (see A177862), so m(6) = 15 = a(6).
%C Subsequence of A177862.
%H O. A. Ivanov, <a href="http://www.ams.org/bookstoregetitem/item=mbk61">Making Mathematics Come to Life: A Guide for Teachers and Students</a>, American Mathematical Society, Providence, RI, 2009; see p. 11.
%H O. A. Ivanov, <a href="http://www.jstor.org/stable/10.4169/000298910x523362">On the number of regions into which n straight lines divide the plane</a>, Amer. Math. Monthly, 117 (2010), 881888.
%e If n = 6, then i = min{j > 0  j*(j+1) >= 2*(61) = 10} = 3, so a(6) = 3*(6+23) = 15.
%t i[n_] := (j = 1; While[j (j + 1) < 2 (n  1), j++]; j); Table[i[n] (n + 2  i[n]), {n, 0, 70}]
%o (PARI) a(n)=if(n<3, n+1, my(i=(sqrtint(8*n8)+1)\2); (n+2i)*i) \\ _Charles R Greathouse IV_, Nov 12 2016
%Y Cf. A000124, A055503, A177862.
%K nonn,easy
%O 0,2
%A _Jonathan Sondow_, Nov 30 2015
