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A263883 Ivanov's number a(n) = i*(n+2-i) where i = min{j > 0 | j*(j+1) >= 2*(n-1)}. 3

%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+2-i) where i = min{j > 0 | j*(j+1) >= 2*(n-1)}.

%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/bookstore-getitem/item=mbk-61">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), 881-888.

%e If n = 6, then i = min{j > 0 | j*(j+1) >= 2*(6-1) = 10} = 3, so a(6) = 3*(6+2-3) = 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*n-8)+1)\2); (n+2-i)*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

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