OFFSET
0,2
COMMENTS
The maximum number of regions into which n lines can divide the plane is A000124(n) = n(n+1)/2 + 1.
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).
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).
Subsequence of A177862.
LINKS
O. A. Ivanov, Making Mathematics Come to Life: A Guide for Teachers and Students, American Mathematical Society, Providence, RI, 2009; see p. 11.
O. A. Ivanov, On the number of regions into which n straight lines divide the plane, Amer. Math. Monthly, 117 (2010), 881-888.
EXAMPLE
If n = 6, then i = min{j > 0 | j*(j+1) >= 2*(6-1) = 10} = 3, so a(6) = 3*(6+2-3) = 15.
MATHEMATICA
i[n_] := (j = 1; While[j (j + 1) < 2 (n - 1), j++]; j); Table[i[n] (n + 2 - i[n]), {n, 0, 70}]
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jonathan Sondow, Nov 30 2015
STATUS
approved