Recurrent Problems - Lines In The Plane ¶
What is the maximum number L<sub>n</sub> of regions defined by lines("unfolding" or "unwinding") in the plane?
small cases ¶
L<sub>0</sub> = 1 L<sub>1</sub> = 2 L<sub>2</sub> = 4 L<sub>3</sub> = 7 .....
===== mathematical expression =====
L<sub>0</sub> = 1, L<sub>n</sub> = L<sub>n - 1</sub> + n, for n > 0
L<sub>0</sub> = 1, L<sub>n</sub> = L<sub>n - 1</sub> + n, for n > 0
closed form ¶
L<sub>n</sub> = L<sub>n - 1</sub> + n
----
= L<sub>n - 2</sub> + (n - 1) + n
= L<sub>n - 3</sub> + (n - 2) + (n - 1) + n
. . . .
= L<sub>0</sub> + 1 + 2 + ... + (n - 2) + (n - 1) + n
= 1 + S<sub>n</sub>, where S<sub>n</sub> = 1+ 2 + 3 + ... + (n - 2) + (n - 1) + n.
= L<sub>n - 3</sub> + (n - 2) + (n - 1) + n
. . . .
= L<sub>0</sub> + 1 + 2 + ... + (n - 2) + (n - 1) + n
= 1 + S<sub>n</sub>, where S<sub>n</sub> = 1+ 2 + 3 + ... + (n - 2) + (n - 1) + n.
S<sub>n</sub> = 1+ 2 + 3 + ... + (n - 2) + (n - 1) + n
+S<sub>n</sub> = n + (n - 1) + (n - 2) + ... + 3 + 2 + 1
2S<sub>n</sub> = (n + 1) + (n + 1) + (n + 1) + ... + (n + 1) + (n + 1) + (n + 1)
S<sub>n</sub> = n(n + 1)/2, for n >= 0
L<sub>n</sub> = n(n + 1)/2 + 1, for n >= 0S<sub>n</sub> = n(n + 1)/2, for n >= 0
