Euler problem 45

Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:

Triangle		Tn=n(n+1)/2		1, 3, 6, 10, 15, ...
Pentagonal		Pn=n(3n1)/2		1, 5, 12, 22, 35, ...
Hexagonal		Hn=n(2n1)		1, 6, 15, 28, 45, ...

It can be verified that T₂₈₅ = P₁₆₅ = H₁₄₃ = 40755.

Find the next triangle number that is also pentagonal and hexagonal.

analyse:
1) Every Hexagonal is also a Triangle.
performance improvements: 
2) Start looping with Hexagonal numbers because they increase faster than other two number types.
3) H143 gives 40755 so start with n = 144.
4) use difference: H(n) - H(n-1) = n(2n-1) - ((n-1)(2(n-1)-1)) = 4 * n - 3

An alternate solution is to make use of the square root formula. x = (-b+- sqrt(b2-4ac))/2a

When we rewrite the pentagon formule P(n) = n(3n-1)/2 

we get 1.5n*n - 0.5n - P(n) = 0 where P(n) = H(n)

Problem45 =  1533776805 elapsed time:    1 ms. Test Passed.