==== 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'''

===== 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.

   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 >= 0'''