Finding how many regions the plane is cut into by k lines that are in general position?

The answer is: (k,0) + (k,1) + (k,2) = (k^2+k+2)/2

where (a,b) is a binomial coefficient.

If k=n(n-1)/2 then

((n(n-1)/2)^2+(n(n-1)/2)+2)/2 = (n^4-2n^3+3n^2-2n+8)/8

giving 1, 2, 7, 18, 56, ...

Ref:
-- Ivan Niven, Mathematics of Choice p.124-126
-- David Wells, Curious and Interesting Puzzles, 464 p.146
-- H. Dorrie, 100 Great Problems of Elementary Math... p.283
-- Martin Gardner, New Mathematical Diversions, p.236, p.242
-- Ross Honsberger, Mathematical Gems I, 12.1 p.134
-- G. Polya, Mathematics and Plausible Reasoning Vol.I, p.47, 3.11 p.54